I'm sorry to say but I don't think there's any way to get rid of the little guys. The actual "body" of the mushroom is underground, where you can't see it. The mushrooms that you see above the ground are just used to make more mushrooms.

The mushrooms will usually go away after some time, but if you need to you can just cut them down with a lawn mower whenever they pop up.

Identification can be difficult with some, so it's best just to leave them alone.