Drywall Patching - Probably easy but i'm a newbie
First you have to measure what thickness it is. It's going to be 5/8 or 1/2.
Go to Lowes or HD and look for the smaller pieces they have for sale, should be 2' X 2'. If they do not have a piece the thickness you need go find a damaged piece of the thickness you need and make a deal with the manager.
You need a framing square, razer knife, some sheetrock screws, small roll of drywall tape, 6" wide drywall knife, sanding sponge, primer and paint and some dry wall compound.
Remove the fan cover, measure what size you need and use the square to lay it out. Just run the knife over the face over the face of it to cut through the paper. Bend the piece back to snap the gypsom and cut it from the back side to cut the paper.
Screw the piece in place making sure the screw is just below the surface but not breaking the paper. Wipe some drywall compound on the seam then set the tape over the compound. Wipe over it lightly to compress most of the compound from under it. Apply another thin coat of compound, the key word being thin.
Let it dry 24 hours, wipe over it to only knock off the high places and go over it with another coat of thin compound and let it dry 24 hours. Now go over it with the sanding sponge, do not over sand down into the paper, if there's any low spots or air bubble just thin a small amount of compound with a few drops of water and go over it again.
No amount of primer or paint will fix any flaws.
Once it's finish sanded prime and paint.