Sally - this pool cover is the polyester kind that is made out of the material that is like shade cloth. The rain will go through it just like it will w/shade cloth (think of the stuff they use on the poles over green house plants in the summer at Meijers, or wherever.) It's not a "solar cover" that's designed to warm up the pool.
I'm not worried about them having cover for rain. They just need shade on part of the run during the day if they're going to be stuck not being able to get out of that pen to range. Otherwise they'd want to be in the house all day to get out of the sun and it would be too close quarters.
I'll have to take a photo of what it looks like tomorrow.
Your pictures are great. It looks wonderful. If you have time and the resources you may want to make some kind of hide aways for them to get to in case of a hawk. It looks like you have plently of bushes and trees for protection when they are free ranging. But if one is sun bathing in a corner it might not be able to run to the coop in time. Although I'm sure you will leave your flags up.