I also have a very small coop, intended only for roosting, and am anticipating this problem as we are expecting a wet month and I have clay soil that drains very slowly and poorly. I have a tarp over my run, which so far has kept most of the run, except the edges, very dry. I put down a thick layer of straw, not to absorb water, but to ensure that the chickens have something other than muck to stand on. So far, so good. I know mold is a possibility, but from advise from others on this site and from talking to folks at local feed stores, I have come to the conclusion that straw is probably my best option. I have no access to pine needles where I live, so that really isn't an option for me.