Check you have enough memory / free space for the songs on your iPhone (Eg an iPhone generally needs around 8Gb free in which to operate smoothly. (NB: If short on space, it is always wise to back up your data before making space) See: How to check the storage on your iPhone and iPad - Apple Support
Check you have a good strong internet connection:
If you have VPN try turning that off.
Check iPhone is well-charged.
You may be able to REdownload the purchased music:
See how to: Redownload music from the iTunes Store on your iPhone or iPad - Apple Support
or
What to do: If you can't connect to the App Store, iTunes Store, or other Apple services - Apple Support
If still no success, you may want to: Contact Apple support