I have to agree with sand as a potential solution. We had a lot of trouble with a muddy run until we put sand in. Then, problem solved. We actually shoveled out a lot of the dirt in the run (beautiful, rich soil for the garden) and then put about 3 to 4 inches of sand in. It worked wonders and now the run only gets damp, no puddles, even where it's not covered. Our run is about 9 x 16 feet and only needed about 2 yards of sand total. And since sand is only about 12.00 a yard here, it was a cheap and relatively easy solution.
If money is not a factor I would scrape off the top layer of organic matter and put down drainage rock followed by wooden horse pellets. I would also install perimeter drains around the outside of the run and coop and ensuring a positive ground slope. I live on the rainy west coast and deal with lots and lots of rain.