Hello Wendy-17,
Thank you for reaching out to Apple Support Communities. We are happy to help you!
We understand that you are having issues with being able to add songs to your music playlist. Have you checked to make sure that the software on the device you are utilizing is up to date? Or have you attempted to do a power cycle on the device?
Nonetheless, we have provided you with a few helpful informational articles:
Make sure that you have Turn on Sync Library with Apple Music.
Add a song to a playlist in Apple Music without adding it to your library
"With Apple Music, when you add a song to a playlist from the Apple Music catalog, you can turn off the option for it to automatically add to your Library. For example, if you create a playlist for holiday music, you can add songs to that playlist, but the songs won't play when you listen to your Library on Shuffle.
Before you change your settings, make sure that you have the latest versions of iOS, iPadOS, macOS, tvOS, and the Apple Music app on Android."
We understand that you are familiar with how to, Add and download music from Apple Music, but this article also provides a few additional helpful steps.
We hope this finds you well!
Cheers!