Start with where the water goes
Every muddy run is water arriving faster than it leaves. So the build starts with siting, not surfacing: put the run on the highest ground available, never in the dip where the garden drains to, and check where roof water from the house, shed or coop currently lands — a run sited under a dripping gutter line will never dry out. Give the ground a gentle fall away from the coop door if you can, so rain sheets off rather than pooling where the hens queue each morning. If the plot is genuinely wet, a shallow drainage channel or a French drain along the uphill side, intercepting water before it enters the run, is worth more than any surface you lay on top of a swamp.
The base layer that does the real work
With water redirected, build the floor. Scrape off the softest topsoil if it is pure clay, lay a permeable weed membrane to stop the layers mixing, and bring in a few inches of coarse hardcore or gravel, compacted, as a free-draining platform. This base is unglamorous and invisible once finished, but it is what stops a run turning to porridge in the first wet month: water passes down through the surface into the base and away, instead of sitting in the top layer with the hens paddling in it. Edge the run with boards or sleepers so your materials stay in and the flock's scratching does not spread them across the lawn.
Woodchip: the working surface
On top of the base goes the surface the chickens actually live on, and deep woodchip is the backyard standard for good reason. A layer several inches deep cushions feet, gives the flock something to scratch through all day, and lets droppings wash down below foot level, so hens walk on chip rather than on their own muck — better for feet, better for smell, better for parasite control. Use coarse, fresh chip rather than fine bark, which mats and slimes. The chip slowly rots down, which is the plan: top it up as the level drops, and when it eventually becomes soil-like, dig it out for the garden and start again. Hens kept permanently on bare earth or concrete need a scratching substrate provided regardless; chip simply builds it into the floor.
A roof over at least part of it
The single most effective mud cure is stopping rain landing at all. Covering part of the run — clear corrugated sheets over the section by the coop, or a tarpaulin on a frame as the budget version — gives the flock a dry area for feeders, drinkers and the dust bath, and halves the water the drainage has to handle. Feeders and drinkers live under the roof on slabs, the dust bath stays dry enough to use, and the hens get an outdoor room they will use in weather that would otherwise pin them in the coop doorway. Leave the rest open for sun and grass access if you rotate ground.
A run build is usually a household project spread over weekends, and the upkeep — chip top-ups, drain checks — stretches over years. Keeping the measurements, the materials list and the maintenance jobs in the flock's shared profile through KinsLe Family Sharing means everyone works to the same plan and the drainage knowledge survives the person who dug it.