Try some sand for the surface. Dawg53 gets sand hauled in each year to make fresh surfaces for his chickens. You can sometimes find it free if near a creek, or buy it at Lowes or Home Depot.
If you could make a slanted roof that overhangs to the outside, that would shunt the water away from them. If you can obtain any used metal or fiberglass roofing panels from someone in the are, you could build up the back side with wood to help slant it. I tried tarps early on, and they were hard to deal with. Plus, it is good to try and make the top predator-proof.