trench around the coop to divert to water and use drainage gravel, its a quick fix for now and you can do something a little more extreme later.  you should make the trench about a foot wide and at least 6 " deep if you can.  digging should be easy for you now with the wet conditions.  worse come to worse you can add onto the roof or put a gutter system on the roof to divert the off flow.
something like this but around a coop
 
if its due to the fact you have a low spot in your yard then you would need to correct that by adding elevation, raise bed or even for temporary purposes you could throw a few pallets under it for now to raise it up off the ground and give them something to walk on that isn't mud.
	
		
			
		
		
	
				
			something like this but around a coop
if its due to the fact you have a low spot in your yard then you would need to correct that by adding elevation, raise bed or even for temporary purposes you could throw a few pallets under it for now to raise it up off the ground and give them something to walk on that isn't mud.
			
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