If you cannot raise the run, you could dig a trench around the perimeter of the run for the water to run away from the chicken yard area. If the trench is filled with gravel, it become a dry river-bed type set-up to direct the water where you want it to go.
We've used plastic pipe as a drain around one side of the barn (the high side) to direct the flow of water to the places WE want the water to go - instead of inside the barn! The pipe is corregated black perferated pipe from the home store - dug in a trench twice as deep as the pipe is round - filled with small 'pea" sized gravel and laid in the middle of the gravel (so the pipe is buried in gravel). Works well.
We attached the other end of the pipe to a dry-well - think large rubbish bin buried underground - the bin is bottom-less and laid in more gravel, so the collected water drains away deep underground instead of sitting on the surface. Don't ask me to shovel any more gravel this year!
By doing this, the water runs away from the chicken run - however, our layers have eaten ALL available greenstuffs to hold the mud at bay. So, come warmer weather, we're constructing 'green boxes' - a 8'x3' raised bed area (think garden bed) likely made out of 2x6" timbers - with hardware cloth stapled over the top of it. Sprinkle in the grass seeds of your choice, and the grass grows through the hardware cloth to be tasty treats for the chickens - but they cannot get to the roots to dig up/kill the plants. I've also thought about re-seeding the yard by limiting their access to certain areas - like a rotating pen idea - but don't favor that concept because I want the chickens to have as much space as possible. If I let them range, the hawks come to watch. That makes me nervous!
I'm hoping to construct 6 or 8 'green boxes' as their run is large - and placing wood chips as mulch so that there isn't as much mud. But please know, you're not the only muddy chicken owner! I don't like rubbing mud off their eggs - because someone had dirty feet in the nestboxes!
