If you have play disc installed, you should be able to play using your scratched disc. The console will check what game it is, then once it sees you have play disc installed and in disc drive it will automatically switch to HDD. Broken disc won't matter by then, so long as it can check the disc is Halo 4
EDIT: IF you're interested, the reason why you cannot play without disc isn't because of a technical problem, it was programmed to not allow you to play with the disc. This is because - think about it. If it was possible to play without it, then a group of 10, 20, 30, 100, 100,000, 100,000,000 people could just buy one game, then share the disc with someone else after they installed it, and pass it on and on and on. They install it, then play it, for free.