The heavy snow last winter cracked the styrofoam and it now sags allowing water to collect and cause more damage. What options are there for repair? Could I use a sheet of plywood glued to the topside of the foam as support? If so what glue works best on styrofoam?

You are on the right track, as long as the plywood (or hardboard) will fit under the cover. 3M makes a good spray adhesive that should work for you. Most home centers will have it. Double check the can to make sure it won't attack styrofoam (polystyrine).