I've dealt with that sort of problem before. You should be able to get two or three screws into the noggin on the left.
On the right I put two pieces of wood behind the plasterboard they were covered in glue so stuck to the back of the plasterboard. The back of those pieces of wood was level with the front of the noggin. A third piece of wood behind them gave me something to screw into. It is easier to do than describe.
Yeah I did think of doing that but that'll prob be a last resort means more filling, sanding & paintingIf you are ok with a bit of finish damage next to the socket another option would be to use screws to fix the wood in place.
It misses the middle part of the box, I can get 2 screws in on the left & that's it.You should be able to get at least 3 screws into the noggin , two near the middle and one at the end , you can drill another hole if you need to.
The only way that would come loose is if you used nails instead of screws , as that noggin covers at least half the back box that would sort the problem , certainly does when I do it
The flanges look good but there hard to source only found them on 2 sitesGet a dry liner box like this
View attachment 199027 PLASTERBOARD FLANGE to fix the right hand side (they are made to solve this problem!)
Get them in lots of places, like this
https://www.tlc-direct.co.uk/Products/AP603.html
Try a local electrical wholesaler, but I see you are in Leicester, so maybe not so easy …?The flanges look good but there hard to source only found them on 2 sites
The right hand side will pull away from the wall over time (as plugs are pulled in and out). So that end needs securely fixing.You should be able to get at least 3 screws into the noggin , two near the middle and one at the end , you can drill another hole if you need to.