Alan, you might be able to take a hacksaw blade or Dremel tool with a cut-off disk in it and cut a new and deep screwdrive slot.
We don't know if this is wood or metal, but they make screw extractors that work best on bolts with a hole drilled in them. Many wood screws are too small for that process but the answer for wood is to get a core drill. The core drill will be a lot harder to find but those take out a cylinder of wood that includes the screw. You then use a dowel to fill the hole. Core drills are available for metal, too, but probably so expensive that it's probably to expensive to consider.