To keep a girl/guy on the side to have sexual intercourse later, because she is not eligible to play for whatever reason . Usually, because she/he is underage by a few months.
T-Shirt as a verb. To t-shirt something is to like it so much that you would want to wear it on the front of a custom t-shirt. This can be an item, photo/image, or phrase/quote.
James: Did you hear about that politician from Chicago?
Amy: Yeah, I bet they'll t-shirt that guy someday.
OR:
He liked his cats so much that he'll be t-shirting their pictures tonight.