Very Simple and very cheap to repair.
1. Go to your local Hardware store and instead of purchasing a full 4x8' drywall ask for there scrap if they have any and find the size you need to cover the hole. Purchase a small bucket of joint compound and some drywall tape.
2. Don't remove the thatching and clean up the edges with a utility knife, sharp one of course, and don't worry about rough edges too much. As long as it don't get bigger and bigger by crumbling.
3. Cut the drywall to fit. When you begin placing the joint compound do just the edges and place in drywall tape over the wet compound, set to dry. Then begin to place in the remaining coats until wall is flush with remaining wall.
Know from the size of the hole there should be studs exposed under the thatching so you can screw the drywall right into it. After the job is done repaint the entire wall so the paint matches completely over the whole wall. Very simple and very cheap, don't lose a security deposit on only a $20 job, and the total labor time is low. so even if you cant do the job don't let the landlord try and scam you out of all the doe.