can't play purchased music

Repeated requests to authorize an already authorized computer for recent store downloads typically indicates you have corrupt or incomplete copies. Delete the tracks that won't play from your library without hiding from your purchase history. Redownload using the cloud links that should remain, or close iTunes and reopen if they don't show up straight away. While downloading click the activity widget and turn off the option that allows for simultaneous downloads.

If that hasn't helped see If iTunes prompts you to authorize your computer when you try to play iTunes Store purchases - Apple Support.

tt2

You can remove and redownload tracks on iPhone. Check the elipsis (three dots) menus.

I don't really know how common this issue is or what factors are involved, although I've a sense that turning off simultaneous downloads improves reliability which is why I recommend it. iTunes has millions of users. I might answer one or two threads on this topic a week.

tt2