Hi 1qeeM,
It sounds like you're having an issue with playing and keeping downloaded content with Apple Music. If so, have you checked your device storage to see iff content is automatically offloading due to a lack of overall storage? The steps below found in the article here can help you check: How to check the storage on your iPhone and iPad - Apple Support
"Use your device to check its storage
- Go to Settings > General > [Device] Storage. You might see a list of recommendations for optimizing your device's storage, followed by a list of installed apps and the amount of storage each one uses.

2. Tap an app's name for more information about its storage. Cached data and temporary data might not be counted as usage.
In the detailed view you can:
- Offload the app, which frees up storage used by the app, but keeps its documents and data.
- Delete the app, which removes the app and its related data.
- Depending on the app, you might be able to delete some of its documents and data.
If your device is almost full and can't free up space, you might get a Storage Almost Full alert. If you see this alert, check the storage recommendations or offload some less-used content like videos and apps."
Also, do you have an Apple Music voice plan? If so, this doesn't allow adding and downloading content from Apple Music. The article here has this information: Add music to iPhone and listen offline - Apple Support
Keep us posted on what you find when checking your storage and confirming your Apple Music plan, or if you have questions about the steps.
Cheers.