What you are seeing underneath the mirror squares is probably the surface of the sheetrock that forms the wall, or, it may be that someone did paint cardboard and somehow attach it to the wall. You can try spackle, but its harder than you might think to get a nice smooth surface using that method unless you are well versed in the art of veneering spackle or plaster onto a wall with a broad knife. It might be easier to re-sheetrock those areas. You can screw new wallboard (another name for sheetrock)right into the existing wall, then use seam tape and joint compound to finish it off, just as was done when the room was built originally. Its important to sand the areas where joint compound is used before priming with a good quality acrylic primer such as Fine Paints of Europe's Eurolux Primer.