Yes, definately need something on top that will shed water. What state are you in? I think you would need to put a solid back and at least one solid end on the coop, instead of just the wire, unless you are in a really hot climate, so the chickens will have some protection from wind, blowing...
Need to get a slant on it. Higher in front than in back, so rain will run off. Perhaps a 2 X 6 set on edge along the front edge, then the plyboard (maybe not a warped piece?) and some tin or shingles on the plyboard to turn water. The slant should make the rain run off. Extend the roof out...