Clay is a problem. It is impervious and does not allow the water to drain away. The particles absorb water and swell up. It's bad enough just with water sitting on it but if you or the chickens walk on it it becomes a real sticky icky mess.
That article linked above is a good article. Pat lived in a swamp in Ontario and knew mud. There are two basic concepts, keep the water out to begin with or get it out once it gets in.
Keeping the water out is usually easier said than done. A roof can help, especially if it is sloped so the water runs away from the run and not into it or use gutters and downspouts to get that water away. Rain and snow can blow in from the side, if you have a prevalent wind direction during storms blocking that side of the run might help. Preventing rainwater runoff from going into your run could help, maybe use berms or swales. But if your run is in a low spot where water drains to it, you usually have issues.
Good ventilation and turning the top can help with evaporation, but it sounds like you are way beyond that. You need a way to drain that water. What you can do will depend a lot on your terrain and the size of your run. If you have a low spot nearby a French drain of some sort may work. In clay it probably will not get rid of all the water but maybe enough.
Some people really like sand, some people say it doesn't work at all. Like any tool, it needs to be used correctly. Sand drains really well if the water has some place to go, but if you put sand in a hole in clay you have a bathtub full of sand and water. I'll mention this here. Chickens poop. If the poop builds up much and it stays wet it can start to stink really bad. The science behind that is that if it is too wet for oxygen to get to the poop the microbes eating it are anaerobic and their digestive byproducts stink. The more chickens you have in a small area the faster the poop builds up. This could be a problem with a bathtub full of sand, water, and poop and is one reason why some people don't like sand.
Pat goes into some detail on doing what you propose. One of the issues is that rock or sand will disappear into the mud over time. It might work better as a dry season project. My personal preference would be to use a round rock like pea gravel or probably that river rock. Crushed gravel is likely to have sharp edges where the chickens might cut their feet when scratching.
Your situation sounds more like Pat's than mine. My coop and run are on a soil that turns into baby excrement when wet but they are sort of on a high spot. I put more dirt into my coop to raise the level even more above the surroundings and that keeps rainwater runoff out. My coop stays dry. I use wood shavings on top. The run drains reasonably well but the holes where they dust bath are a real mess when it is wet. I've dumped pea gravel outside the human door between the coop and run to give some better footing but it disappears over time. I occasionally have a few large scraps of lumber to give dry places.
There are temporary solutions. Dumping sand or gravel in the coop or run can help. pallets or scrap lumber to walk on can get you and them out of the mud. Various types of bedding can really help. You may be able to leave it or you may need to remove it, it can work both ways.
Good luck. Dealing with mud is no fun.
