I purchased a song says not authorized can you help

There should be a button that has 3 dots. Press that and there should be an option that says "remove..." press that and hit the "remove downloads" button and re download it. It should now work!!

Welcome to the Apple Community.

Try deleting the problematic file (electing to remove original file if/when prompted) and then re-downloading the file from the iTunes store.

You can re-download content purchased from the iTunes store (availability varies depending on location) using the purchased option from the Quick Links section in the top right corner of the iTunes homepage in your iTunes application on your computer.

You can re-download content purchased from the iTunes store (availability varies depending on location) using the purchased option at the bottom of the screen of the iTunes app (or video app) on your iOS device.

If the problem re-occurs, select the content which is causing a problem and use the 'Report a problem' button in Your Purchase History using your computer.

I had this happen to me today, delete the songs from your library (not a playlist such as "purchased" and then re-download them by going to the playlist "Purchased" and clicking on the "Download Previous Purchases" in the bottom right.

This page will help: http://support.apple.com/kb/HT2519

Realize a response to a topic that dates from 5 years ago may be with solutions that didn't apply back then. 😉