You can put a rubber band around the circumference of a clay target, then use fishing line to tie the rubber band to your backboard. You can then use the rubber band to hold the next clay target, unless your shot cut the rubber band.
We did this at the club, but hung them from a target frame, and used fishing line stretched behind the clays to keep them from turning in the wind. Its amazing how few times we had to replace any of the fishing line.