HCM III ACQUISITION CORP

Exhibit 10.9

HCM III ACQUISITION CORP

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Stamford, CT 06902

[●], 2025

HCM Investor Holdings III, LLC

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Suite 330

Stamford, CT 06902

Ladies and Gentlemen:

This letter will confirm our agreement that, commencing on the effective date (the “Effective Date”) of the registration statement on Form S-1 (the “Registration Statement”) for the initial public offering (the “IPO”) of the securities of HCM III Acquisition Corp, a Cayman Islands exempted company (the “Company”), and continuing until the earlier of (i) the consummation by the Company of an initial business combination and (ii) the Company’s liquidation (in each case, as described in the Registration Statement) (such earlier date hereinafter referred to as the “Termination Date”), HCM Investor Holdings III, LLC, a Delaware limited liability company (the “Sponsor”), shall take steps directly or indirectly to make available to the Company, at ▇▇▇ ▇▇▇▇▇ ▇▇▇▇▇▇▇▇ ▇▇▇▇▇, ▇▇▇▇▇ ▇▇▇, ▇▇▇▇▇▇▇▇, ▇▇▇▇▇▇▇▇▇▇▇ ▇▇▇▇▇ (or any successor location), office space and secretarial and administrative services as may be required by the Company from time to time. In exchange therefor, the Company shall pay the Sponsor a sum of up to $15,000 per month on the Effective Date and continuing monthly thereafter until the Termination Date. The Sponsor hereby agrees that it does not have any right, title, interest or claim of any kind (a “Claim”) in or to any monies that may be set aside in a trust account that may be established upon the consummation of the IPO (the “Trust Account”) and hereby irrevocably waives any Claim it may have in the future as a result of, or arising out of, any negotiations, contracts or agreements with the Company and will not seek recourse against the Trust Account for any reason whatsoever.

This letter agreement constitutes the entire agreement and understanding of the parties hereto in respect of the subject matter hereof 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.

The parties may not assign this letter agreement or any of their respective rights, interests or obligations hereunder without the consent 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 shall be governed by, construed in accordance with and interpreted pursuant to the laws of the State of New York.

This letter agreement may be executed in one or more counterparts, each of which shall for all purposes be deemed to be an original but all of which together shall constitute one and the same agreement. Only one such counterpart signed by the party against whom enforceability is sought needs to be produced to evidence the existence of this letter agreement.

[Signature Page Follows]

	Sincerely,
	HCM III ACQUISITION CORP
	By:
		Name:	▇▇▇▇▇ ▇▇▇▇▇▇▇▇
		Title:	Chief Financial Officer

AGREED AND ACCEPTED BY:
HCM INVESTOR HOLDINGS III, LLC
By:
	Name:	▇▇▇▇▇ ▇▇▇▇▇▇▇▇
	Title:	Managing Member

[Signature Page to Administrative Support Agreement—HCM III Acquisition Corp]