Hi alicja52,
Thanks for using Apple Support Communities! I see you're having an issue with playing songs on your iPhone with Apple Music, and I'd like to do what I can to help.
Have you restarted your iPhone since noticing this issue? If not, go ahead and do that, as a restart can resolve unusual issues.
Restart your iPhoneÂ
If you're still having the same issue, does the same thing happen when connected via Wi-Fi?
If not, you may need to enable Cellular Data at Settings > Music.
If you'd prefer to not use cellular data with Apple Music, you can download music for listening offline, and leave cellular data turned off. Add music to iPhone and listen offline
Take care!