I don't know if you'll be able to have two different type of content (a game + an app) at the same time, handled by the Wii U.
BUT, it's nearly 100% sure, and a bit of fresh info: in the actual dev kits, a large chunk of the hardware memory is reserved by Nintendo for the OS, for multi-tasking, for the possibility to put your game on hold and go to the OS, run an application, etc. The game is "stored", and you can go back to it instantly. There are applications running in backward also (the internet ones i guess, etc.). Because of this, third-parties have a known memory pool of X GB in dev kit, but Y GB of it is not available for their games (and i don't talk about debugging here).
So in the end i think there are chances that if the game allows it (read: it doesn't need a dual screen setup, so think of a VC Gamecube title), you can play it on one screen, and the other is running a not-ressources-intensive app. At the very least, the hardware is capable of that.