A possible way to fix your problem. Drill a large hole centered in the middle of those four screws. At least an inch, larger if possible to hide it with the end of the handle. Work a 1" x 1"x X" piece of wood into the hole. The depth of the wall will decide how long that piece can be and still be able to slide it into the wall. Once in the wall place it where the existing screws will go into it and run a small flat head screw through the thin luan between 2 of your existing screw holes to hold it it place. Repeat with another piece of wood for the other 2 screw holes. Now you have some solid wood to screw the handle to and they are supported by a larger area of the thin wood.