I've had similar problems since winter hit. The weather is never good enough to get it to dry out.
My response has been two fold:
I) I've put weed suppressant fabric down and then covered with a good layer of hardwood chips (don't use bark mulch or similar). I found a local supplier for the ones from bark-uk. The chickens tend to spread it around and pile it up, but its easy to rake back flat again.
ii) I've put a roof on the run that Ill remove in spring/summer. I just used pvcu corrugated sheets from Wickes.
Between the two, it is far far better than it was. Before, it was a bit like Glastonbury, but now the mud is solid and the eggs cleaner. There is also a very dry area which serves as a great dustbath.