If it is plasterboard on a wooden studwork frame, cut away the wallpaper with a craft knife, beyond the damage. Remove the architrave at the top.
Cut away the damaged plasterboard to expose the studwork that you are going to fix to.
Measure the thickness of the old plasterboard.
Procure some plasterboard and cut a rectangle big enough to replace the damage and fix to the studwork
Hold it up to the wall and draw a pencil mark round it
Cut along the pencil mark and cut out the rectangle.
You should now have a hole that your repair piece will fit into neatly
This is much easier that making a hole and trying to cut a patch that will fit.
Drill holes and fix the patch with countersunk screws, not nails, driving the screws slightly below the surface.
Use "joint cement" or "jointing compound" and a broad metal knife to fill the joints and hide the screws.
Use the minimum amount, pressed down very hard, and scrape off any excess before it sets
Do not overfill it and leave lumps that you will have to sand off later.
Examples
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