Cut a rectangle of screen 1 inch larger than the damaged area. It should be at least 2-inches-by-2-inches. Purchase window screen at a hardware store.
Cut a 36-inch length of thread. Thread a needle with the thread and knot the ends of the thread together. If you are using upholstery thread. Use 24 inches and knot only one end. Upholstery thread is thicker so you do not have to double it. Place a piece of clear tape over the knot to keep the thread from pulling through the screen.
Pin the patch to the screen with straight pins.
Sew around the edge of the patch 1/4 inch from the edge. Make your stitches 1/4-inch long. Stitch by running your needle through the holes of the screen; piercing the fibers of the screen will damage it.
Sew all the way around the patch, overlapping the beginning of the seam by 1 inch. Cut your thread 1 inch from the screen. This is a temporary fix so you do not need to try to knot the ends of the thread.