Don't know what your run looks like, a picture could expand suggestions.
PVC or repurposed gutters can make a quick fix for water diversion.
If the water comes in or runs through at specific locations, you can employ permaculture techniques. Such as, digging a swale, adding a berm, or making a channel. Channels can be lined with stones that allow water to drain while chicken feets can still utilize the space it takes up.
Very temporary solutions can include laying some boards down, then it's up to chickens to choose to stand on the boards or the mud.
I've got sympathy. My run isn't completed yet, because my ducks are in the way, as they're still learning to free range. Boy do ducks know how to make mud! For duck challenges in particular:
Placing water fount on top of a large pavers stone (2.5' square?) So they cannot do the back and forth water, dirt, water, dirt deal.
Shavings clean out from their house gets spread on ground, giving them at least 70% of the run not mud.
Limiting access to the swimming water. We have one of those black rubbery cattle feed(?) Bowls. It's big, gotta be 15 gallons? Temporary until we can get a duck pond more suitable. Blocking all but one access side to this was critical in preventing them from billing all of the water onto ground. Filling water 4 times a day was too much for me!