First, read this.
Pat’s Big Ol' Mud Page (fixing muddy runs):
https://www.backyardchickens.com/articles/how-to-fix-a-muddy-run-chicken-coop.47807/
Mud is wet clay. Clay holds water, does not let it drain away like sand does. Mud is not much fun. It is messy and can be hard to walk in. It sounds like that is part of your problem. But organic matter like their poop and a lot of types of bedding decomposes, rots, composts, breaks down (whatever you want to call it) when the microbes that rots it can get enough water to live and reproduce. If the moisture level is low enough that they can breathe air then aerobic (air breathing) microbes do the job. These are what you want. If it is too wet for them to breathe air the anaerobic microbes take over. These produce a strong smell and can get really slimy. Pretty disgusting. This sounds like a big part of your problem.
There are two basic solutions, it usually takes a combination. You can try to keep the water out to start with. Where is the water coming from? If it is groundwater runoff then landscaping such as berms and swales may be part of the answer. Are nearby roofs channeling rainwater in there? The solution may be gutters and downspouts or sloping the roof so it drains somewhere else. A roof over the run can help, especially if the water off of it flows somewhere else, but rain and snow can blow in form the sides. While a roof is usually a big part of the solution the run will still get wet in a rain if you have any wind. If your run is in a low spot where water drains to it this becomes much harder.
The other part is to get the water out once it gets in. This gets a little more tricky. Evaporation can help but if more water is coming in this is not solution, it takes too long. Water will drain downhill if it has a place to go and a way to get there. Sand drains very well, clay does not. Soil type has a lot to play in this. Putting sand on top of clay is often not the simple solution if you are in a low spot. If you dig a hole in clay and fill it with sand the water has no place to go. The anaerobic microbes will still make a stink and the sand will eventually disappear into the mud. You have to take into account your soil types, landscape, and drainage patterns. There is no one solution that fits everybody.
Sometimes putting something on top of the mud can work, either as a temporary solution or even long term. That might be putting pallets, boards, something for them and you to walk on. It might be some type of mulch. Some people use mulch as a temporary solution and have to dig it out later but others find it a good long term solution.
Pat talks about adding a layer of rock to keep the sand from sinking into the mud so fast, then topping it with sand. If you do that I'd consider building up the run with more clay to get it above the surrounding area before putting the rock on. That gives the water a place to drain to. It can be a lot of work, especially with a large run, but if done right it can provide excellent drainage.
If you have a nearby low spot to drain to some type of French drain can move a lot of water. That could be perforated pipe, gravel, or sand.
If you clay is a fairly shallow layer and you have a layer of well drained soil like sand under it, digging some holes straight down, like with post hole diggers, and filling those holes with sand or small gravel can give the water a place to go.
I don't know what the right solution for you is, there are too many variables. For my ground level coop I added a few inches of clay to raise the impervious level above the surrounding ground and put a berm and swale on the uphill side. It stays bone dry in there.
My 12' x 32' main run is sort of on the top of a slight rise so it drains reasonably well. It still has holes where they dust bath in dry weather and turn to mud when wet. The berm and swale protect the uphill side of that. It is covered which helps a lot but it still gets pretty wet when rain sets in. It is a silty clay so it can get really muddy if I or the chickens walk on it. I dumped a few bags of pea gravel to build up and firm up areas where I walk to keep the mess down. I have about 3,000 square feet of grass inside an electric fence which is where they spend most of their time so the poop doesn't build up in the main run that much. Even when it stays wet a few days the smell isn't bad at all. If they were enclosed in a small place the poop could build up a lot more and would probably stink.
We all have different circumstances so we don't all have the same problems or the same solutions. Good luck.
