I had the same problem after purchasing an album through iTunes. The only thing I found that worked was a complete iPhone Restore.
This is done through iTunes on the device summary tab. There's a button just below the "Check for Update" button that is titled: "Restore". MAKE SURE YOU SYNC FIRST so that everything is backed up. After the restore is complete iTunes will automatically re-install all your info (contacts, calendars, apps, etc...) from your last backup.
Genius - thanks very much, that worked. Even without the music sync'd to the phone, those mystery tracks that couldn't be played were still there and couldn't be deleted (what the heck does that circle icon mean, anyway?!).
Restoring the phone and those songs are gone. I don't know if they'll come back, but it's fixed it for now - thanks very much for your help
i wondered the same thing.
You have to go to the "Restrictions" page of your device's Settings and make sure "iTunes" is turned on.
This is why it only happened on my children's iPods and not on mine.
The reason a full restore works is that it resets all your passwords and allows everything automatically until you reset the restrictions again.
How curious :)
I can't test this anymore because - ah hah - I'd done the full restore job. I don't know why restrictions would have been enabled on my phone, I hadn't turned them on - but good to know for next time.
(I'm guessing you're referring to the Restrictions page that's about parental controls... Which makes sense as you're talking about your kids' devices).