Hey Yawhen,
Thanks for using Apple Support Communities! I understand you're running into an issue enabling explicit content options with Apple Music.
Checking that out in Screen Time restrictions is a great first step. Are you able to make the changes in Settings > Screen Time > Content & Privacy Restrictions > Content Restrictions > Music, Podcasts & News? Are you either already showing "Explicit" there, or able to select it?
If Explicit is already enabled, or the issue continues after enabled that, try going to Settings > Screen Time and then all the way to the bottom there, tap to turn off Screen Time, restart your device, and see if you're able to access explicit content in Apple Music. If you are, go ahead and enable Screen Time again test once more. Let us know how that goes.
Use Screen Time on your iPhone, iPad, or iPod touch
Restart your iPhone
Thanks again, and have a great day!