Exhibit 10.4

Hennessy Capital Acquisition Corp. III
3485 N. Pines Way, Suite 110

Wilson, Wyoming 83014

June 22, 2017

Hennessy Capital LLC
3485 N. Pines Way, Suite 110

Wilson, Wyoming 83014

        Re: Administrative Services Agreement

Gentlemen:

This letter will confirm our agreement that, commencing on the date the securities of Hennessy Acquisition Corp. III (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 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”), Hennessy Capital LLC (“Hennessy Capital”), an affiliate of our sponsor, Hennessy Capital Partners III LLC, shall make available to the Company, at 3485 N. Pines Way, Suite 110, Wilson, Wyoming 83014 (or any successor location), certain office space, utilities, and general office, receptionist and secretarial 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 and will be entitled to be reimbursed for any out-of-pocket expenses until the Termination Date.

Hennessy Capital hereby irrevocably waives any and all right, title, interest, causes of action and claims of any kind (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”), 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.

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 all 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, 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 laws principles.

[Signature page follows]

AGREED TO AND ACCEPTED BY:

HENNESSY CAPITAL LLC

[Signature Page to Administrative Services Agreement]