It's probably cross threaded. You can find a screw removal tool for your power drill, reverse the drill. Drill a small starting depression in the stuck screw, then put the tip of the removal bit into the depression. It should back the screw out. The only problem will be if the threads in the base are damaged, you might not be able to put a new screw in it.
Lowes has an OK selection of fine screws, but you'll have to match threads exactly. Or order replacement screws from the manufacturer. You didn't say what brand router. But many in that price range, such as the venerable Bosch 1617, have a removable base, so you can change it out between fixed and plunge base. You can order a new base for the damaged one.
When drilling holes for mounting, it's a good idea to use the technique illustrated. A slightly oversized main hole and a larger, shallow hole on top of the first deep enough to lower the head of the screw below the surface. The oversized main opening allows for a little error. I always use a center punch before drilling. Tape the base down. Many plate makers make the holes pre drilled for most routers.
