Read this. It has things I’ll miss. Maybe one or two of them will strike a spark for you. I’ll repeat some things others have said.
Pat’s Big Ol' Mud Page (fixing muddy runs):
There are two basic concepts, keep water out and, if it gets in, get it out.
Put berms or swales on the upslope side to divert groundwater. Your roof does need a slope, water that pools on top will either seep through or rot a wooden roof. Slope it so the water runs away from the run or use downspouts and gutters to direct it away.
A roof is good but wind blows rain in from the sides too. Covering the direction the rain comes from is a huge benefit. You can use tarps, that 3 mil clear plastic, or anything that stops water from getting in. That will increase wind area so you may need to do some additional bracing. Putting a diagonal from one top corner to the opposite bottom corner of an open space does a whole lot of good. It’s generally not the weight of the siding that’s the problem, it’s the wind.
Then you have the problem of getting the water out when it gets in. When it sets in wet like you are seeing it’s next to impossible to keep all the water out. Gravity pulls water downhill but it has to have a place to drain to. Sand is permeable, water drains through it really well. Clay is not, it will hold water forever. If you dig a hole in clay and fil it with sand, all you’ve done is create a bathtub filled with sand. The water won’t go anywhere. Sand is great for removing water, but the water has to have a place to go.
There are a few issues with sand. Eventually it will disappear down into the mud. Its density is greater than clay so gravity pulls it down. The chickens scratching doesn’t help. Not only does their scratching help mix it with the clay and get it to sink, they will scratch a lot of it through the fence. Eventually you’ll need to top off the sand. You can help yourself by putting down a layer of gravel on top of the clay and under the sand. Don’t dig the clay out, put the gravel on top of the clay. That really helps stop sand from disappearing downward. And put some type of barrier around the bottom of your run to stop them from scratching the sand out. A heavy rain can wash it out but that barrier helps with that too.
I know I said put sides up to keep rain out, but once it quits raining evaporation really helps it dry. Maybe like Blooie said, keep one end open or remove some of that siding after the rain stops.
The water needs a low place to drain to. Your slope probably provides that. But a “French drain” from that run to a low spot may be greatly beneficial. That French drain might involve a pipe with holes in it, but just a ditch backfilled with gravel or even coarse sand will move a lot of water.
Chickens need a place to get their feet dry. Walking in mud and water some won’t hurt them but if their feet are constantly wet that can cause problems. You can put things in there for them to stand on, cinder blocks, timbers or planks, plywood, a lot of different stuff. I’d also be a little nervous about most plastic as it might be slick.
Another option is to scatter straw, hay, wood shavings, wood chips, things that give them a surface to get above the water. People do that and it works, but there are some potential issues with it. When you have a compost pile you want it damp so the aerobic bugs can eat that stuff and turn it into compost. Aerobic bugs give it a nice earthy smell. But if it stays soaking wet anaerobic bugs (bugs that don’t breathe oxygen) take over. They’ll still break it down but the compost is lower quality and it becomes slimy and stinky. Whoo, man does it stink! A lot of people compost in the run, keep stuff in there all the time and it works great, but occasionally if it stays too wet some people have to dig this stuff out and replace it in the run. Everything is not always clear cut. What works for some won’t necessarily work for everyone.
Your stinky problem is poop. Poop is like that compost pile. If it is dry or just barely damp it’s fine, no smell. But when it gets wet it starts to stink. Chickens poop all the time. The more chickens you have in a small space the more the poop builds up. No matter what kind of floor you have in your run, they are going to poop in it. Even with sand that drains well you might get a whiff but it’s nothing like a run that holds water. Some people with sand in their run are out there every day scooping up clumps of poop to help with this. Some of us with clay floors or other run flooring never clean them out and don’t have any real problems. We are all unique. Different things work for different people.
I don’t know it any of this will help you or not. I hope you get some benefit out of it. When it sets in wet it can get tough in many ways.