How do I fix drywall around wall box so switch plate covers damage?
I wouldn't start cutting out chunks of drywall to fix a small area like that.

Take off the wall plate and remove any loose or crumbling drywall.  Apply a piece of fiberglass tape over the area, mud, sand and repeat as necessary to get a blended finish. Then prime and paint to match.

The fiberglass tape is nice for repairs like this because you can stick it to the wall and then mud on top of, unlike paper tape which you need to apply on top of mud.  I've fixed a bunch of similar defects like this and several years later there are no cracks and they look great.