EX-10.5





 Exhibit 10.5 

HUDSON EXECUTIVE INVESTMENT CORP. III 

570 Lexington Avenue, 35th Floor 

New York, NY 10022 

February 26, 2021 
 Hudson Executive Capital
LP 570 Lexington Avenue, 35th Floor New York, NY 10022 
 HEC Sponsor II LLC 570 Lexington Avenue, 35th Floor New York, NY 10022 

Re: Administrative Services Agreement 

Ladies and Gentlemen: 
 This letter agreement
(this “Agreement”) by and among Hudson Executive Investment Corp. III (the “Company”), HEC Sponsor III LLC (the “Sponsor”) and Hudson Executive Capital LP, an affiliate of the
Sponsor (“HEC”), 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”): 
 1. HEC shall make available, or cause to be made available, to the Company, at 570 Lexington Avenue, 35th Floor, New
York, NY 10022 (or any successor location), office space and secretarial and administrative services as may be reasonably required by the Company. In exchange therefor, the Company shall pay HEC $10,000 per month on the Listing Date and continuing
monthly thereafter until the Termination Date; and 
 2. The Sponsor and HEC each 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 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”), and hereby
irrevocably waives any Claim it may have in the future as a result of, or arising out of, this Agreement, 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. 

  
 1 













 This 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 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 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 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. 

[Signature Page Follows] 












 
			
	Very truly yours,
	
	HUDSON EXECUTIVE INVESTMENT CORP. III
		
	By:	 	 /s/ Jonathan Dobres

		 	Name: Jonathan Dobres
		 	Title: Chief Financial Officer


  

			
	AGREED AND ACCEPTED BY:
	
	HUDSON EXECUTIVE CAPITAL LP
		
	By:	 	 /s/ Douglas G. Bergeron

		 	Name: Douglas G. Bergeron
		 	Title: Managing Partner
	
	HEC SPONSOR III LLC
		
	By:	 	 /s/ Douglas G. Bergeron

		 	Name: Douglas G. Bergeron
		 	Title: Managing Partner

 [Signature Page to Administrative Services Agreement]