Exhibit 10.4


 


HENNESSY CAPITAL INVESTMENT CORP. VI


3415 N. Pines Way, Suite 204


Wilson, Wyoming 83014


 


September 28, 2021


 


Hennessy Capital Group LLC


3415 N. Pines Way, Suite 204


Wilson, Wyoming 83014


 


Re: Administrative Support Agreement


 


Ladies and Gentlemen:


 


This letter agreement by and
between Hennessy Capital Investment Corp. VI (the “Company”) and Hennessy Capital Group LLC (“Hennessy Capital”),
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 Global 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) Hennessy Capital shall
make available, or cause to be made available, to the Company, at 3415 N. Pines Way, Suite 204, Wilson, Wyoming 83014 (or any successor
location of Hennessy Capital), certain office space, utilities and secretarial and administrative support as may be reasonably required
by the Company. In exchange therefor, the Company shall pay Hennessy Capital the sum of $15,000 per month on the Listing Date and continuing
monthly thereafter until the Termination Date; and


 


(ii) Hennessy Capital
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]


 




     


     


    

 


	 	Very truly yours,
	 	 
	 	HENNESSY CAPITAL INVESTMENT CORP. VI
	 	 	 
	 	By:	 /s/ Daniel J. Hennessy
	 	 	Name:  	Daniel J. Hennessy
	 	 	Title: 	Chief Executive Officer


 


	AGREED TO AND ACCEPTED BY:	 
	 	 
	HENNESSY CAPITAL GROUP LLC	 
	 	 
	By:	 /s/ Daniel J. Hennessy	 
	Name:	Daniel J. Hennessy	 
	Title:	Managing Member	 


 


[Signature Page to Administrative Support Agreement]