When you are in check during a game of chess can you checkmate while not protecting your king?

Yes. The only way is to move a piece so that it blocks your king from check and checks or attacks another piece of the opponent. Another way is for the king itself to attack the piece that is attacking it. Example is if the queen foolishly moves adjacent to the king without being protected by another piece, the king may capture the queen. It is not limited to simply moving out of check.