Help New User - Push Mail on ICS


Hi there,

I have a HTC One S running Android 4.0.3 and HTC Sense 4.0.
I'm trying to get push email (not polling) working on this phone.
I've only had it about a week and can't quite figure out what is happening.
After reading about the place I discovered K-9 Mail which seems to work.
Ok, it does work, but my phone seems to become dormant or something.
If I'm using the phone I can receive what I preceive to be push email.
When I lock the screen the emails seem to keep coming through.
Once the screen has been locked for a fair while it seems the pushability ends.
It's like the phone is no longer contactable.
This holds true not just for notifications too.
I'll get all notifications whilst actually using the phone and for a short time after locking the screen but after a time those will just stop outright too.
Any ideas what is going on?!