Harry Caray was an iconic figure in Chicago sports history. He was famous for his play-by-play work as a White Sox announcer. His trademarks included his "one, two, three, strikes you're out" phrase and the use of a baseball bat to point to a batter's number.
Harry was also well known for his singing during commercial breaks. He would hold the microphone out to fans as a way of thanking them for their support.
The word harry carry meaning comes from the Japanese term hara-kiri. It is basically the kun'yomi of seppuku. Seppuku is an act of ritual suicide that involves disemboweling yourself with a sword. In casual use hara-kiri has become a slang term for suicide in general. The phrase is often heard in American media, particularly film and television from the 1940s to 1960s era.
In the World War II era propaganda movie Across the Pacific, the character Dr. Lorenz attempts hara-kiri when his plot to sabotage the Panama Canal is foiled by Humphrey Bogart's Rick Leland. However, he fails to complete the act and is captured instead. In the same vein, a failure to commit to one's responsibilities and duties might be referred to as a harry carry.