No
Answer
Adding re to listen would produce the word 'relisten'. Perhaps you could say 'I am relistening to the news' or 'Please relisten to the messages on the answering machine'. However, most people would, most correctly, say 'I am listening to the news again' or 'Please listen to the messages on the answering machine again'. Re and again mean the same thing, but the latter is used as the 'correct' version.