Improve control over which parts of the audio the character follows

I just did my first Hedra with a song lip sync

And noticed two things that bothered me:

1. when the music (not vocals)started, the actors lips seemed to react to it and that didn't look right.

2. If there are back up vocals, the main actor tries to sing them as well. There needs to be a way to fix these in post. I only want my actor to sing what's intended for him to sing, and not be all over the place trying to sing every part. Thank you,