Hello elusivezoomie,
Are you receiving the verification code as expected when attempting to sign in on the app?
The following steps for restarting the app may help:
1. From the Home screen, tap Apps.
2. Tap Settings.
3. Scroll to Apps, then tap it.
4. Tap Apple Music.
5. Tap Force Stop. If they see a message that says "If you force stop an app, it may misbehave," tap OK.
6. Return to the Home screen or the application list, and reopen Apple Music.
7. If the issue persists, restart the device.
If restarting the app hasn't helped then clearing the app's cache may also help:
1. From the Home screen, tap Apps.
2. Tap Settings.
3. Scroll to Apps, then tap it.
4. Tap Apple Music.
5. Tap Storage, then tap Clear Cache.
6. Return to the Home screen or the application list, and reopen Apple Music.
If you continue to have issues then you may try reaching out to the Apple Support Advisors that specialize in that app here: Apple Support
Cheers.