My run was once nice and grassy but they have since turned it into dirt. With the rain and stuff It’s kind of hard to avoid the mud. Granted the chain link fence isn’t very water tight at the bottom and hopefully the new run that’s framed out with an actual roof will help when it comes to keeping it dry
Sounds like you will need to divert the water from the edges of your run and coop. Lots of ways to do that - berm and swale, trenches, french drain, gutters on a roofed run & possibly others (?). Look around BYC, I'm don't have good advice. Your chickens can and may dig down at the base of your run (the wooden edging). This can allow chickens out, water and predators in. You just shift it back or bury some type of footing in your run that goes below ground. The ground aprons will keep predators from getting in and you can keep it deep in DLM. You can always move material around to refill holes.
Our coop/pens are open air due to where we are (heat/humidity more of an issue than snow/cold) in the sandhills of NC. We have sand. A lot of sand. I hate sand. We have had multiple varieties of wild life, ponies, horses, chickens, ducks, dogs and cats in our 7 acre pasture area. We have had multiple wildlife (some are predators, some not), chickens, ducks, cats and dogs in the yard areas circling the house. I don't know all the types of livestock that has been run off and on through the years on this property - I do know that parts have been cleared in the past and allowed to re-grow into the forested areas we currently have. The feces of any animal never completely gets picked up no matter how you scoop or rake. Then, if that sand gets wet (it can in open air coops like ours), it smells BAD.
So, we are actively trying to change that sand into usable loamy material that will grow grasses, weeds and crops for livestock AND can also be used for gardening in beds directly on the ground, raised beds and hugelmounds. Our chickens help with that by their natural digging, scratching and of course, manuring. In the stationary coop/pens, we use DLM (Deep Litter Method). There are many posts on that here on BYC, I will add a couple to get you started.
I use a lot of different products. I have added in shavings - both fine and coarse - at different times, but not regularly. So, I have both long & short needle pine straw, leaves of various types (mostly oak, but fruit trees starting to drop now, too), weeds, lawn clippings when we mow, shredded paper/cardboard/junk mail, garden trimmings, garden veggies & fruits, trimmings from veggies/fruits we eat, hay and straw, wood chip mulch. As these products break down, they encourage bug life and allow the chickens to scratch down and dig. As it breaks down, we add more. You can tell if we've missed any coop/pens - if it smells or is actively muddy. When it builds up enough that I feel i can remove some to use elsewhere, I do. I never remove all of it as you want it to stay and start new materials to composting down. I am rebuilding our chicken coop/pens so that they can hold up to 10-12" of materials deep at a time to break down. It can take a lot of natural material to layer that deep, LOL...
Our
DLM photo album.
DLM on BYC -
ChrisnTiff(2007) ,
Mac (11/11/2007) . Yes, there are other threads out there, but to me these have the most info and a LOT of pictures showing how to do, how to use etc. You can then search BYC from there for the many, many threads on DLM.
Our chicken tractors don't get bedding. Instead they get moved. Depending on the size of the tractor, the number of & type of chickens in it and how old they are - sometimes we wait a week, sometimes every other day and sometimes as much as 2x daily (morn and afternoon feeding). I TRY not to let the birds scratch and dig so much that they create huge holes or even completely destroy the grass but sometimes that happens. Either way, the areas they have been - grow fast and a lot. We deal with the weeds (no chemicals here) by putting the chickens back over them if possible and by mowing when we have to. After 2 years of multiple tractors in our back yard, side yard and front yard - we couldn't keep up with the mowing this past fall - to some of our friends who believe in only lawns - it was "awful", LOL. Have a lot of growth that actually froze down or was knocked down in the extremely wet weather we've been having. That's ok too - green manure for growth next spring.

I have trees that were thought to be dead/dying that suddenly in 2020 (we moved in in 2015 - I never thought about established trees in a yard needing to be "fed". I knew nothing of trees), grew like CRAZY due to the fertilization from the moving chicken tractors. In some that's awesome, in others... they need to go now. Working on that, too.