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and recess the penny washers slightly
Do you use some sort of auger-like drill bit to recess, or just screw them up tight so they pull and recess themselves?
I hadn't thought about doing it myself as it was previously lifting. But tbh going to screwfix and doing it the way you've described actually sounds easier than finding someone and arranging for them to do it.
Is there anything else I should check for?
Probably easier and more effective to get some penny washers and plasterboard screws. Screw the board back to the joist. It's sometimes enough to do that and recess the penny washers slightly then fill/decorate.
You have to judge whether it would work based on the condition of the plasterboard.