Apple Music keeps prompting me to authorize your computer when you try to play purchases.

Repeated requests to authorize an already authorized computer for store downloads typically indicates you have corrupt or incomplete copies. Backup and then 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/Music 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.

tt2

So you’ve authorized the computer, and it hasn’t been accepted?

Try rebooting then, and see if that clears up the issue.

If that fails, deauthorize all, and then re-auth those that you’re using.

(Above assumes the purchases aren’t tied to another Apple ID.)

If your songs and apps and the rest are are associated with a different Apple ID, or are otherwise seemingly corrupted, delete them all and re-load them. (Changing an Apple ID is a bigger effort than might be realized.)

See also Authorization vs. association - Apple Community for the complications that can arise from trying to access past purchases from multiple IDs.

tt2