Document:

EX-10.4

 Exhibit 10.4 

HAYMAKER ACQUISITION CORP. III 

501 Madison Avenue, Floor 12 

New York, NY 10022 

March 1, 2021 
 Mistral Capital Management
LLC 
 501 Madison Avenue, Floor 12 
 New York, NY 10022 

Re: Administrative Support Agreement 

Ladies and Gentlemen: 
 This letter agreement by
and between Haymaker Acquisition Corp. III (the “Company”) and Mistral Capital Management LLC (“Mistral”), dated as of the date hereof, will confirm our agreement that, commencing on the date the securities of the
Company are first listed on The Nasdaq Capital Market (the “Listing Date”), pursuant to a Registration Statement on Form S-1 and prospectus filed with the U.S. Securities and Exchange
Commission (the “Registration Statement”) and continuing until the earlier of the consummation by the Company of an initial business combination or the Company’s liquidation (in each case as described in the Registration
Statement) (such earlier date hereinafter referred to as the “Termination Date”): 
 (i) Mistral shall make available, or
cause to be made available, to the Company, at 501 Madison Avenue, Floor 12, New York, NY 10022 (or any successor location of Mistral), certain office space, utilities and secretarial and administrative support as may be reasonably required by the
Company. In exchange therefor, the Company shall pay Mistral the sum of $20,000 per month on the Listing Date and continuing monthly thereafter until the Termination Date; and 

(ii) Mistral hereby irrevocably waives any and all right, title, interest, causes of action and claims of any kind as a result of, or
arising out of, this letter agreement (each, a “Claim”) in or to, and any and all right to seek payment of any amounts due to it out of, the trust account established for the benefit of the public stockholders of the Company and
into which substantially all of the proceeds of the Company’s initial public offering will be deposited (the “Trust Account”) as a result of, or arising out of, this letter agreement, and hereby irrevocably waives any Claim it
may have in the future, which Claim would reduce, encumber or otherwise adversely affect the Trust Account or any monies or other assets in the Trust Account, and further agrees not to seek recourse, reimbursement, payment or satisfaction of any
Claim against the Trust Account or any monies or other assets in the Trust Account for any reason whatsoever. 
 This letter agreement
constitutes the entire agreement and understanding of the parties hereto in respect of its subject matter and supersedes all prior understandings, agreements, or representations by or among the parties hereto, written or oral, to the extent they
relate in any way to the subject matter hereof or the transactions contemplated hereby. 
 This letter agreement may not be amended,
modified or waived as to any particular provision, except by a written instrument executed by the parties hereto. 
 No party hereto may
assign either this letter agreement or any of its rights, interests, or obligations hereunder without the prior written approval of the other party. Any purported assignment in violation of this paragraph shall be void and ineffectual and shall not
operate to transfer or assign any interest or title to the purported assignee. 
 This letter agreement constitutes the entire relationship
of the parties hereto, and any litigation between the parties (whether grounded in contract, tort, statute, law or equity) shall be governed by, construed in accordance with, and interpreted pursuant to the laws of the State of New York, without
giving effect to its choice of law principles. 
 [Signature Page Follows] 

  
 1 

 
			
	Very truly yours,
	
	HAYMAKER ACQUISITION CORP. III
		
	By:	 	 /s/ Steven J. Heyer

		 	Name: Steven J. Heyer
		 	Title: Chief Executive Officer

  

			
	AGREED TO AND ACCEPTED BY:
	
	MISTRAL CAPITAL MANAGEMENT LLC
		
	By:	 	 /s/ Christopher Bradley

	Name:	 	Christopher Bradley
	Title:	 	Managing Member

 [Signature Page to Administrative Support Agreement]EX-10.5

 Exhibit 10.5 

PRIVATE PLACEMENT

WARRANTS PURCHASE AGREEMENT 

THIS PRIVATE PLACEMENT WARRANTS PURCHASE AGREEMENT, dated as of March 1, 2021 (this “Agreement”), is entered into by and
between Haymaker Acquisition Corp. III, a Delaware corporation (the “Company”), and Haymaker Sponsor III LLC, a Delaware limited liability company (the “Purchaser”). 

WHEREAS, the Company intends to consummate an initial public offering of the Company’s units (the “Public Offering”),
each unit consisting of one share of the Company’s Class A common stock, par value $0.0001 per share (a “Share”), and one-fourth of one redeemable warrant, each whole warrant
exercisable for one Share at an exercise price of $11.50 per Share, as set forth in the Company’s registration statement on Form S-1 related to the Public Offering (the “Registration
Statement”); and 
 WHEREAS, the Purchaser now wishes to purchase an aggregate of 5,333,333 warrants (or 5,933,333 warrants if the
underwriters’ over-allotment option is exercised in full) (the “Warrants”), each Warrant entitling the holder to purchase one Share at an exercise price of $11.50 per Share. 

NOW THEREFORE, in consideration of the mutual promises contained in this Agreement and other good and valuable consideration, the receipt and
sufficiency of which are hereby acknowledged, the parties to this Agreement hereby, intending legally to be bound, agree as follows: 

AGREEMENT 

Section 1. Authorization, Purchase and Sale; Terms of the Warrants. 

A. Authorization of the Warrants. The Company has duly authorized the issuance and sale of the Warrants to the Purchaser. 

B. Purchase and Sale of the Warrants. 

(i) As payment in full for the 5,333,333 Warrants being purchased under this Agreement, the Purchaser shall pay $8,000,000
(the “Purchase Price”), by wire transfer of immediately available funds in accordance with the Company’s wiring instructions, at least one (1) business day prior to the effective date of the Registration Statement, or on
such other date as the Company and the Purchaser may agree. 
 (ii) In the event that the underwriters’
over-allotment option is exercised in full, the Purchaser shall purchase up to an additional 600,000 Warrants (the “Additional Warrants”), in the same proportion as the amount of the over-allotment option that is exercised, and
simultaneously with such purchase of Additional Warrants, as payment in full for the Additional Warrants being purchased hereunder, and at least one (1) business day prior to the closing of all or any portion of the over-allotment option, or on
such other date as the Company and the Purchaser may agree, the Purchaser shall pay $1.50 per Additional Warrant, up to an aggregate amount of $900,000, by wire transfer of immediately available funds in accordance with the Company’s
wiring instructions. 
 (iii) The closing of the purchase and sale of the Warrants shall take place simultaneously with
the closing of the Public Offering (the “Initial Closing Date”). The closing of the purchase and sale of the Additional Warrants, if applicable, shall take place simultaneously with the closing of all or any portion of the
over-allotment option (such closing date, together with the Initial Closing Date, the “Closing Dates” and each, a “Closing Date”). The closing of the purchase and sale of each of the Warrants and the Additional
Warrants shall take place at the offices of Ellenoff Grossman & Schole LLP, 1345 Avenue of the Americas, New York, New York 10105, or such other place as may be agreed upon by the parties hereto. 

 C. Terms of the Warrants. 

(i) The Warrants shall have their terms set forth in a Warrant Agreement to be entered into by the Company and a warrant
agent, in connection with the Public Offering (a “Warrant Agreement”). 
 (ii) At or prior to the time
of the Initial Closing Date, the Company and the Purchaser shall enter into a registration rights agreement (the “Registration Rights Agreement”) pursuant to which the Company will grant certain registration rights to the Purchaser
relating to the Warrants and the Shares underlying the Warrants. 
 Section 2. Representations and Warranties of the Company. As
a material inducement to the Purchaser to enter into this Agreement and purchase the Warrants, the Company hereby represents and warrants to the Purchaser (which representations and warranties shall survive the Closing Dates) that: 

A. Organization and Corporate Power. The Company is a corporation duly organized, validly existing and in good standing
under the laws of the State of Delaware and is qualified to do business in every jurisdiction in which the failure to so qualify would reasonably be expected to have a material adverse effect on the financial condition, operating results or assets
of the Company. The Company possesses all requisite corporate power and authority necessary to carry out the transactions contemplated by this Agreement and the Warrant Agreement. 

B. Authorization; No Breach. 

(i) The execution, delivery and performance of this Agreement and the Warrants have been duly authorized by the Company as
of the Closing Dates. This Agreement constitutes a valid and binding obligation of the Company, enforceable in accordance with its terms. Upon issuance in accordance with, and payment pursuant to, the terms of the Warrant Agreement and this
Agreement, the Warrants will constitute valid and binding obligations of the Company, enforceable in accordance with their terms as of the Closing Dates. 

(ii) The execution and delivery by the Company of this Agreement and the Warrants, the issuance and sale of the Warrants,
the issuance of the Shares upon exercise of the Warrants and the fulfillment of, and compliance with, the respective terms hereof and thereof by the Company, do not and will not as of the Closing Dates (a) conflict with or result in a breach of
the terms, conditions or provisions of, (b) constitute a default under, (c) result in the creation of any lien, security interest, charge or encumbrance upon the Company’s capital stock or assets under, (d) result in a violation
of, or (e) require any authorization, consent, approval, exemption or other action by or notice or declaration to, or filing with, any court or administrative or governmental body or agency pursuant to the certificate of incorporation or the
bylaws of the Company (in effect on the date hereof or as may be amended prior to completion of the contemplated Public Offering), or any material law, statute, rule or regulation to which the Company is subject, or any agreement, order,
judgment or decree to which the Company is subject, except for any filings required after the date hereof under federal or state securities laws. 

C. Title to Securities. Upon issuance in accordance with, and payment pursuant to, the terms hereof and the Warrant
Agreement, the Shares issuable upon exercise of the Warrants will be duly and validly issued, fully paid and nonassessable. Upon issuance in accordance with, and payment pursuant to, the terms hereof and the Warrant Agreement, the Purchaser will
have good title to the Warrants and the Shares issuable upon exercise of such Warrants, free and clear of all liens, claims and encumbrances of any kind, other than (i) transfer restrictions hereunder and under the other agreements contemplated
hereby, (ii) transfer restrictions under federal and state securities laws, and (iii) liens, claims or encumbrances imposed due to the actions of the Purchaser. 

D. Governmental Consents. No permit, consent, approval or authorization of, or declaration to or filing with, any
governmental authority is required in connection with the execution, delivery and performance by the Company of this Agreement or the consummation by the Company of any other transactions contemplated hereby. 

 E. Regulation D Qualification. Neither the Company nor, to its
knowledge, any of its affiliates, officers, directors or beneficial stockholders of 20% or more of its outstanding securities, has experienced a disqualifying event as enumerated pursuant to Rule 506(d) of Regulation D under the Securities
Act of 1933, as amended (the “Securities Act”). 
 Section 3. Representations and Warranties of the Purchaser.
As a material inducement to the Company to enter into this Agreement and issue and sell the Warrants to the Purchaser, the Purchaser hereby represents and warrants to the Company (which representations and warranties shall survive the Closing
Dates) that: 
 A. Organization and Requisite Authority. The Purchaser possesses all requisite power and authority
necessary to carry out the transactions contemplated by this Agreement. 
 B. Authorization; No Breach. 

(i) This Agreement constitutes a valid and binding obligation of the Purchaser, enforceable in accordance with its terms,
subject to bankruptcy, insolvency, fraudulent conveyance, reorganization, moratorium and other laws of general applicability relating to or affecting creditors’ rights and to general equitable principles (whether considered in a proceeding in
equity or law). 
 (ii) The execution and delivery by the Purchaser of this Agreement and the fulfillment of and
compliance with the terms hereof by the Purchaser does not and shall not as of the Closing Dates conflict with or result in a breach by the Purchaser of the terms, conditions or provisions of any agreement, instrument, order, judgment or decree to
which the Purchaser is subject. 
 C. Investment Representations. 

(i) The Purchaser is acquiring the Warrants and, upon exercise of the Warrants, the Shares issuable upon such exercise
(collectively, the “Securities”), for the Purchaser’s own account, for investment purposes only and not with a view towards, or for resale in connection with, any public sale or distribution thereof. 

(ii) The Purchaser is an “accredited investor” as such term is defined in Rule 501(a)(3) of Regulation
D under the Securities Act and the Purchaser has not experienced a disqualifying event as enumerated pursuant to Rule 506(d) of Regulation D under the Securities Act. 

(iii) The Purchaser understands that the Securities are being offered and will be sold to it in reliance on specific
exemptions from the registration requirements of the United States federal and state securities laws and that the Company is relying upon the truth and accuracy of, and the Purchaser’s compliance with, the representations and warranties of the
Purchaser set forth herein in order to determine the availability of such exemptions and the eligibility of the Purchaser to acquire such Securities. 

(iv) The Purchaser did not enter into this Agreement as a result of any general solicitation or general advertising within
the meaning of Rule 502(c) under the Securities Act. 
 (v) The Purchaser has been furnished with all
materials relating to the business, finances and operations of the Company and materials relating to the offer and sale of the Securities which have been requested by the Purchaser. The Purchaser has been afforded the opportunity to ask questions of
the executive officers and directors of the Company. The Purchaser understands that its investment in the Securities involves a high degree of risk and it has sought such accounting, legal and tax advice as it has considered necessary to make an
informed investment decision with respect to the acquisition of the Securities. 

 (vi) The Purchaser understands that no United States federal or state
agency or any other government or governmental agency has passed on or made any recommendation or endorsement of the Securities or the fairness or suitability of the investment in the Securities by the Purchaser nor have such authorities passed upon
or endorsed the merits of the offering of the Securities. 
 (vii) The Purchaser understands that: (a) the
Securities have not been and are not being registered under the Securities Act or any state securities laws, and may not be offered for sale, sold, assigned or transferred unless (1) subsequently registered thereunder or (2) sold in
reliance on an exemption therefrom; and (b) except as specifically set forth in the Registration Rights Agreement, neither the Company nor any other person is under any obligation to register the Securities under the Securities Act or any state
securities laws or to comply with the terms and conditions of any exemption thereunder. 
 (viii) The Purchaser has such
knowledge and experience in financial and business matters, knowledge of the high degree of risk associated with investments in the securities of companies in the development stage such as the Company, is capable of evaluating the merits and risks
of an investment in the Securities and is able to bear the economic risk of an investment in the Securities in the amount contemplated hereunder for an indefinite period of time. The Purchaser has adequate means of providing for its current
financial needs and contingencies and will have no current or anticipated future needs for liquidity which would be jeopardized by the investment in the Securities. The Purchaser can afford a complete loss of its investments in the Securities. 

Section 4. Conditions of the Purchaser’s Obligations. The obligations of the Purchaser to purchase and pay for the Warrants
are subject to the fulfillment, on or before the Closing Dates, of each of the following conditions: 
 A. Representations
and Warranties. The representations and warranties of the Company contained in Section 2 shall be true and correct at and as of the Closing Dates as though then made. 

B. Performance. The Company shall have performed and complied with all agreements, obligations and conditions contained
in this Agreement that are required to be performed or complied with by it on or before the Closing Dates. 
 C. No
Injunction. No litigation, statute, rule, regulation, executive order, decree, ruling or injunction shall have been enacted, entered, promulgated or endorsed by or in any court or governmental authority of competent jurisdiction or any
self-regulatory organization having authority over the matters contemplated hereby, which prohibits the consummation of any of the transactions contemplated by this Agreement or the Warrant Agreement. 

D. Warrant Agreement. The Company shall have entered into a Warrant Agreement with a warrant agent on terms satisfactory
to the Purchaser (the “Warrant Agreement”). 
 Section 5. Conditions of the Company’s Obligations. The
obligations of the Company to the Purchaser under this Agreement are subject to the fulfillment, on or before the Closing Dates, of each of the following conditions: 

A. Representations and Warranties. The representations and warranties of the Purchaser contained
in Section 3 shall be true and correct at and as of the Closing Dates as though then made. 

B. Performance. The Purchaser shall have performed and complied with all agreements, obligations and conditions
contained in this Agreement that are required to be performed or complied with by the Purchaser on or before the Closing Dates. 

C. No Injunction. No litigation, statute, rule, regulation, executive order, decree, ruling or injunction shall have
been enacted, entered, promulgated or endorsed by or in any court or governmental authority of competent jurisdiction or any self-regulatory organization having authority over the matters contemplated hereby, which prohibits the consummation of any
of the transactions contemplated by this Agreement or the Warrant Agreement. 

 D. Warrant Agreement. The Company shall have entered into the Warrant
Agreement. 
 Section 6. Termination. This Agreement may be terminated at any time after June 30, 2021 upon the election by
either the Company or a Purchaser entitled to purchase a majority of the Warrants upon written notice to the other parties if the closing of the Public Offering does not occur prior to such date. 

Section 7. Survival of Representations and Warranties. All of the representations and warranties contained herein shall survive the
Closing Dates. 
 Section 8. Definitions. Terms used but not otherwise defined in this Agreement shall have the meaning assigned
to such terms in the Registration Statement. 
 Section 9. Miscellaneous. 

A. Successors and Assigns. Except as otherwise expressly provided herein, all covenants and agreements contained in this
Agreement by or on behalf of any of the parties hereto shall bind and inure to the benefit of the respective successors of the parties hereto whether so expressed or not. Notwithstanding the foregoing or anything to the contrary herein, the parties
may not assign this Agreement, other than assignments by the Purchaser to affiliates thereof. 
 B. Severability.
Whenever possible, each provision of this Agreement shall be interpreted in such manner as to be effective and valid under applicable law, but if any provision of this Agreement is held to be prohibited by or invalid under applicable law, such
provision shall be ineffective only to the extent of such prohibition or invalidity, without invalidating the remainder of this Agreement. 

C. Counterparts. This Agreement may be executed simultaneously in two or more counterparts, none of which need contain
the signatures of more than one party, but all such counterparts taken together shall constitute one and the same agreement. 

D. Descriptive Headings; Interpretation. The descriptive headings of this Agreement are inserted for convenience only
and do not constitute a substantive part of this Agreement. The use of the word “including” in this Agreement shall be by way of example rather than by limitation. 

E. Governing Law. This Agreement and the rights and obligations of the parties hereunder shall be construed in
accordance with and governed by the laws of New York applicable to contracts wholly performed within the borders of such state, without giving effect to the conflict of law principles thereof. The parties hereto irrevocably submit to the exclusive
jurisdiction of any federal court sitting in the Southern District of New York or any state court located in New York County, State of New York, over any suit, action or proceeding arising out of or relating to this Agreement. To the fullest extent
they may effectively do so under applicable law, the parties hereto irrevocably waive and agree not to assert, by way of motion, as a defense or otherwise, any claim that they are not subject to the jurisdiction of any such court, any objection that
they may now or hereafter have to the laying of the venue of any such suit, action or proceeding brought in any such court and any claim that any such suit, action or proceeding brought in any such court has been brought in an inconvenient forum.

 F. Amendments. This letter agreement may not be amended, modified or waived as to any particular provision, except
by a written instrument executed by all parties hereto. 
 [Signature page follows] 

 IN WITNESS WHEREOF, the parties hereto have executed this Agreement to be effective
as of the date first set forth above. 
  

			
	COMPANY:
	
	HAYMAKER ACQUISITION CORP. III
		
	By:	 	 /s/ Christopher Bradley

		 	Name: Christopher Bradley
		 	Title: Chief Financial Officer

  

			
	HAYMAKER SPONSOR III LLC
		
	By:	 	 /s/ Steven J. Heyer

		 	Name: Steven J. Heyer
		 	Title: Managing Member

 [Signature Page to Private Placement Warrants Purchase Agreement]

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