How to fix a hole above an outlet box?
You should turn off the breaker feeding that receptacle just because you will be taking the cover off.

What you do is get a new small piece of sheetrock, but larger than the hole, cut into a rectangle with nice clean edges. Then trace the outline of that new piece over the hole and using a "keyhole Saw", cut out the wall in that shape, so that the new piece fits in snugly. Then with the piece out, get some small pieces of thin wood, like lath / furring strips or even the old paint stirring sticks (but not plastic ones) and using construction adhesive like Liquid Nails, glue those sticks to the BACK of the wall so they are across the hole.* Let that dry and harden over night. Then put the cut piece back in the hole so that the wood slats are keeping it from pushing through, and use wallboard tape and mud to fix it in place and blend the texture back into the wall. Wait a few days for everything to dry before repainting.

https://www.lowes.com/projects/repair-and-maintain/patch-and-repair-drywall/project


If you CAN, make the new hole go over the 2x4 stud a little, then you have a more solid anchor for that new piece.