UNITED STATES

SECURITIES AND EXCHANGE COMMISSION

WASHINGTON, D.C. 20549

FORM 6-K

REPORT OF FOREIGN PRIVATE ISSUER

Pursuant to Rule 13a-16 or 15d-16

under the Securities Exchange Act of 1934

For the month of May 2026

COMMISSION FILE NUMBER: 001-33373

CAPITAL CLEAN ENERGY CARRIERS CORP.

(Translation of registrant’s name into English)

3 Iassonos Street

Piraeus, 18537 Greece

(Address of principal executive offices)

Indicate by check mark whether the Registrant files or will file annual reports under cover of Form 20-F or Form 40-F.

Form 20-F ☒ Form 40-F ☐

Attached as Exhibit I hereto is a copy of the press release of Capital Clean Energy Carriers Corp. announcing the financial results for the first quarter ended March 31, 2026.

SIGNATURES

Pursuant to the requirements of the Securities Exchange Act of 1934, the registrant has duly caused this report to be signed on its behalf by the undersigned, thereunto duly authorized.

Dated: May 11, 2026

CAPITAL CLEAN ENERGY CARRIERS CORP.

/s/ Gerasimos (Jerry) Kalogiratos

Name: Gerasimos (Jerry) Kalogiratos

Title: Chief Executive Officer

Capital Clean Energy Carriers Corp. 6-K