Why are you having mud issues? Does the area not drain properly? Run off from coop or other roofs? Your first and most important task is to assess the drainage problem and fix that. Poor drainage will result in anaerobic soil that will be a haven for pathogens.
Perhaps installing drainage ditches, tiles or pea stone, adding a gutter system to any roofs involved will dry out the area.
Then, you can address the run. I would turn it into a deep litter run, as others have suggested. If you can get wood chips delivered, that would be fantastic. But, you won't be able to have aged chips delivered. While it's best to be putting aged chips into your run, I'd not be adverse to using fresh chips if that's all you can get your hands on. A combination of wood chips, leaves, grass clippings, garden debris, hay, old coop litter will build a nice composting deep litter. The DL will be a haven for beneficial bacteria and fungi which will keep pathogens in check, help your flock to build a healthy gut flora for improved digestion and immunity, and be a food source for beneficial worms and insects which will ALSO feed your chickens. The DL will act as a sponge, sopping up excess moisture during wet times and releasing it during times of drought.
IMO, bare soil is abused soil. Soil was created to have a covering over it, whether it be vegetative or mulch of one sort or an other.
