Hi, yes you will need to put a sheet of ply down on top of the existing floor baords as this will strenghten the floor up and prevent it form moving, which will then stop the tiles from moving and coming away.
you can use 4 mm ply, but i would use at least 6mm as this will be stronger, but when you lay it, it doesn't have to be 100% perfect to the edge of the walls and is best to leave about a 10mm gap all the way around between the edge of the ply and the wall.
when fixing it down, i would use screws no longer than 20mm as you could be in danger of hitting pipe work below and make sure you put loads of screws in, i.e 1 screw about every 15 cms apart all over the floor as this will stop the baords from bulging in any areas making the tiles come loose. more screws the better.
make sure you buy flexible floor adhesive as this will help with any movement that the floor has, as there will always be some movement.
hope this helps.