"connect to itunes to use push notifications"

Push notifications rely, in part, upon your phone provider(e.g.-Apple routes messages via AT&T, in the US). That message will appear on un-officially unlocked iphones. You say your phone is not jailbroken, but is it unlocked?

Try getting a new sim card from AT&T, I had to ask the question about hacking the phone, as that error message will occur on hacked phones. Since your phone is not hacked, I'm suspecting a problem with your account or a bad sim. So, contact AT&T and ask for a new sim card. If that doesn't fix things, post back.

FYI: In the US, Apple handles all warranty & support issues for the iphone, not AT&T. In your case, AT&T has sim cards though, not Apple.

The problem is because you have restored your iPhone from a previous backup that contained old app data. The old app data was restored and it contains data from a previous version of that app that didn't have push notification capabilities. When you upgraded to the new app, the old app data didn't upgrade correctly so you see this warning to connect to iTunes.

I too once had this problem and the only fix was to restore my iPhone as a new phone. Once I did that and installed the "problem" app, push notifications worked correctly for the app.

I've been using my iPod Touch for ~2 years with no issues. It is not altered in any way. This problem started after upgrading to iOS 9. After restoring to factory and reinstalling my apps, the issue started again.

It's an iOS 9 issue. When someone in Apple admits the problem and fixes it, it will go away.

My bad, I said "iOS 9" in my previous reply, but I meant "iOS 4". Sorry for the confusion.