Greetings Danaphile,
Thanks for visiting the Apple Support Communities! It seems you're encountering issues playing music with your iPod touch, as you mentioned the song will revert to 0:00 during playback, or kick you out of the app.
If you're an Apple Music member, I suggest ensuring you're signed in to the iTunes Store with the correct Apple ID, and that Sync Library is enabled: Sign in to the App Store and iTunes Store on your iPhone, iPad, iPod touch, Mac, PC, or Apple TV & Turn on Sync Library with Apple Music
If this behavior is happening with streaming music, then I suggest testing the behavior while your iPod is connected with a different network, and testing the behavior after downloading the music for offline listening: Add and download music from Apple Music or the iTunes Store
If you're an Apple Music member, and if the music is downloaded for offline listening, then let's have you remove the download while keeping the music in your library, and test the behavior: Delete music, movies, and TV shows from your device
If this is occurring with music that you've synced from iTunes, then let's have you connect your iPod to your computer with iTunes, select your iPod within iTunes, select "Music" under "Settings" in the sidebar, uncheck "Sync Music" and click "Apply" to remove the music. Then check "Sync Music" and "Apply" once again to sync the music back to your iPod and test the behavior: Sync your iPhone, iPad, or iPod using iTunes on your computer
Let me know how it goes from here.
Kind regards.