First question to ask about a wet run is if water drains to it or away from it? Ideally, it is situated on an elevated site and water drains away.
Second issue.........not difficult to understand the source of the smell.....chicken droppings mixed with mud. Dealing with the droppings, which are nigh in nitrogen is the key. A lot of folks use sand, but keep in mind, this is only small rock.....a hard mineral. What is happening is any of the droppings are simply washing through and collecting down below at the soil level. Key word being "collecting". The problem is quite literally building up on you.
So a good plan is deep litter, which more or less acts like a disposable diaper. With a high carbon, coarse deep litter, the N in the droppings start to combine with the C in the litter, which along with the water and air will start to decompose the litter. The best of these I've found is coarse grass hay.....starting about 6 inches deep and building as needed. A foot or so is not too much. With that, the birds are isolated from the mud and mess down deep below......and a slow composting process starts. Over time, this flips from being a liability to an asset, with what develops being some pretty good stuff to add to gardens, flower beds, etc.
Flat wood chips (like pine chips sold for bedding), grain straw, etc. tend to NOT be good for this, as they tend to pack and mat down, blocking out the air. Those tend to turn into wet, smelly mess too.
Coarse, waste type grass hay tends to work best. Legume hay, like alfalfa, clover, etc. are not. They contain the N and also a lot of leaf, that will mat down. Coarse grass hay....and make it deep.
