An application that's supposed to have "push" is not working..

Delete the app and download it again, and or make sure "push notifications" are on in settings.

 

I know nothing about that app, but my last resort is to erase it from the phone and, instead of buying it in the mobile store, re-download it on a computer and transfer it over using USB.

And if that doesn't work you might just have to give up (or at least wait for an update).

 

This happens more often with updates to large games than to smaller programs, but I guess it can affect any app. I don't know if it's a developer mistake or an iOS problem, but there's still apparently something different between buying ON the phone vs. syncing an app TO the phone.

Go figure.