Did president harry truman Regret his decision to authorise the atomic bombings of Hiroshima and Nagasaki? And why did he authorise the bombings?

No, Thuman never regretted his decision. He authorized it wanting to prevent massive casualties on both sides in the planned invasion of Japan: Kyūshū was to be invaded in November 1945. and Honshū four months later. He thought Japan would not surrender unless there was an overwhelming demonstration of destructive capability.