I need to fix this slightly pushed in tile and get it looking normal again. Can anybody advise on what I should do, e.g. what to buy and general instructions/technique? I'm worried that it's going to fall through! , and . Thanks!
As Steven mentions in comments you could have rot in the wall. I wouldn't say this is the case but it is definitely a possibility.
You need to take the tile out. Get a putty knife or something comparable and chisel out the grout. Try to pry it off. If that doesn't work then get a wood block and tap the crap out of the block on top of the tile with a hammer. Every 20 or so taps test tile to see if it is loose. You should give pretty firm taps. The tile will come loose this way or push into the wall if wall is wet.
From there you have to either fix the wall issue and reinstall or just reinstall with more material. You might need a little thinset for it to be pushed out or layering mastic on mastic.