How to repair hole in plasterboard wall?

You need to put a wooden plate on the brackets attached to the back of the timber Then you need to remove the plasterboard so that it enters the hole, and screw it to the previously installed wood, secure the edges with tape for joint filling and for the best effect, smooth it 3 times

You can use a jointing tape and filler and sand to a smooth finish and paint over but I find that you can usually still see the outline of the patch with a keen eye, particularly if the area is brightly lit or near to a window. I usually find that the the best method when rectifying patches, is to plaster the whole wall but it depends on the area, size of the wall and your expectations.

To get a nice smooth look once it's been filled over fibre tape you will need to really feather out the filler spreading it maybe 10 inches otherwise you may be left with an unsightly square frame of filler on the wall, you can find out how to feather filler in on YouTube quite easily

What the builders have done so far sounds correct. Personally because the area is quite large it would be more beneficial to have a patch repair in plaster (skim over repaired area). Then painted would would look good as new.

I would repair the patch of plaster by just redoing the whole wall then paint.

if you have a spare piece of plasterboard cut it a bit bigger than the hole put it over the hole mark the new piece on the wall cut off the marked plasterboard, an easy one is to use to some plywood or wood board screw it in place put the new piece over, screw it on the timber, easy from here the tape over joint , filler follow with paint and done.