Exhibit 10.4

HENNESSY CAPITAL INVESTMENT CORP. VIII

195 US Hwy 50, Suite 207

Zephyr Cove, Nevada 89448

February 4, 2026

Hennessy Capital Group LLC

195 US Hwy 50, Suite 207

Zephyr Cove, Nevada 89448

Re: Administrative Support Agreement

Ladies and Gentlemen:

This letter agreement by and between Hennessy Capital Investment Corp. VIII (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 195 US Hwy 50, Suite 207, Zephyr Cove, Nevada 89448 (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 shareholders 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. VIII

	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]

Hennessy Capital Investment Corp. VIII 8-K