Q: How do I get music purchased on my phone onto my iTunes?

Greetings antonjeffes,

Thanks for the question. If I understand correctly, iTunes repeatedly asks you to authorize even though you already are authorized. I would recommend that you read this article, it may be able to help the issue.

If iTunes repeatedly prompts you to authorize your computer - Apple Support

OS X

1. Quit iTunes and any other open applications until only Finder is open.
2. From the menu bar at the top of your screen, choose Go > Go to Folder.
3. Type /Users/Shared/SC Info into the window.
4. Click Go.
5. If Finder is not already set to column view, set it by choosing View > as Columns from the menu bar at the top of your screen.
6. Find the SC Info folder. Click and hold on it, and drag it to the trash.
7. Restart your computer.
8. Open iTunes.
9. Authorize your computer. If you have multiple accounts or Apple IDs, authorize your computer for each account.

Thanks for using Apple Support Communities.
Take care