Many American CV's were SMASHED by IJN Kamikazes during WW II. Here are some pics of the damage. No Essex class CV was sunk by kamakazies during the war.
The USS Franklin CV-13 was so severly damaged it returned to the US and was totally re-fit. However it never was placed back in service and was laid-up in reserve until 1959.
1) CV-11 Damaged in action on 30 October 1944. View showing work progress on 3 November 1944 at 1230 – flight deck at Fr. 125. (Ship's Photo No. 1003, taken 3 November 1944.) Puget Sound Navy Yard photo # 3984-44.
2) CV-13 damage>> View on the flight deck, looking forward, while the carrier was in New York Harbor, circa 28 April 1945. She had just returned from the Pacific for repair of battle damage received off Japan on 19 March 1945. Note damage to her flight deck, large U.S. ensign flying from her island, and the Manhattan skyline in the background. Official U.S. Navy Photograph, now in the collections of the National Archives (photo # 80-G-K-4760).
3) USS Franklin (CV-13) burning off the Japanese coast after she was hit by air attack, 19 March 1945. Photographed from USS Santa Fe (CL-60), which was alongside to help with firefighting and rescue work. Official U.S. Navy Photograph, now in the collections of the National Archives (photo # 80-G-273888).
4) CV-14 damage.>> Puget Sound Navy Yard image # 943-45. Damage in action of 21 January 1945. Looking stbd. showing damage to superstructure caused by Hit #2.
Seattle Branch of the National Archives photo. Record group 181.
5) CV-17 damage>>Aircraft wreckage on the flight deck, after most fires were out following hits by two Kamikazes off Okinawa, 11 May 1945. Official U.S. Navy Photograph, now in the collections of the National Archives (# 80-G-259904).