Push the wire back in
Before you fix the protruding wire, push it back into the bra so that it isn't sticking out. Dab a little bit of superglue to hold the wire back in and keep the fabric from allowing the wire to stick out again.
Cut a long piece of moleskin (about an inch wide) and peel it off. Then, stick it over the wire and the fabric where the wire was protruding. You can dab a little bit of superglue underneath the moleskin to help it stay put. Repeat on the other bra cup if necessary.
Problem solved — you can go back to being comfortable while wearing your bra.