how do I get my purchased music to play on my iMac?

Repeated requests to authorize an already authorized computer for 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 for Windows prompts you to authorize your computer when you try to play purchases - Apple Support.

tt2

Highlight a song or album and tap a play icon. Assuming that isn't working for you what happens when you try? Are there any error messages you can share?

tt2