iTunes on Phone damaged in move to new phone

When I got a new phone the store moved all my data to the new phone. My iTunes library, which is also stored on my mac mini, was damaged in the transfer. Now it has the wrong album cover for most of the songs, and almost all of the songs are on the phone twice. To play an album, each song will play twice.

I have an undamaged copy of the whole library on my mac, yet when I sync the phone the issues with the iTunes library on the phone are not fixed.

How do I reload the music library from the computer to the phone without messing up the library on the mac or damaging anything else on the phone?