Assuming the run sits on a site that is reasonably well drained, best suggestion I can offer is a deep bed of organic matter. Hay, straw, wood chips, etc. At least 6 inches or so to start......maybe deeper if it's loose and fluffy like hay.
I put that down on a garden area next to my coop.......a place where they hang out all the time. It has all the manure......but you seldom notice it, as each rain takes it down to the soil layer below where it combines with the bottom layer of the hay, moisture, air to start breaking down the hay into rich organic matter. As it rots down, I add more on top. Maybe one bale of hay a month. The grass hay tends to have a stronger stem, so doesn't collapse and mat down as much as baled wheat straw does when it gets wet. Legume hay like clover or alfalfa wouldn't last as long.
Wood chips you can get from powerline cutting crews would also be good. Your area may also have a municipal mulching site for ground up wood mulch. That would be good too.
For the short run, you just keep building it up.....a foot or more of soft, spongy, mostly dry on top bedding. Long term.......a year or so......if it gets too deep, you can go in and clean most of it out and start over. The stuff you take out goes on a garden, flower beds, around trees, etc.