Michelle,
The old screws have probably been torn out of the wood, widening the holes and have nothing to grip onto any more.
Take one of those screws with you to B&Q (or Homebase, etc) and buy the same length screw - but the next size (or two) up in width - if you compare the original with those in the store and it looks like a 3mm screw - buy a small pack of 4mm or 5mm width screws.
These fatter screws will grip in the enlarged hole in the wood frame.
Note: You may need to gently enlarge the original screw holes in the table support to let the fatter screws pass through.
Note: You may want to dip the original screws in wood filler (B&Q) and try to pack that into the wood - this will set into new "wood" and strengthen the wood - just before you use the new fat screws
Note: Using a two-part Epoxy adhesive will help hold the table support to the wall boarding, but that's not a total solution to the problem.
Robert