Have your sim click on the chicken. You should see an option to tell the chicken to "Preen" itself. Click on that. 
Then check the coop to make sure it's clean.
Thank you for replying! Your helpful suggestions are hugely appreciated.
Hmm... I don't see the "Preen" option anywhere. I've included a few more screenshots of various sections of the interaction menu. That is all I see available... Admittedly, this could be because one or more of my 300+ mods are interfering with the Preen option's ability to show up. I shall investigate this further in my game, and thank you again for pointing that out.
And yes, I clean the coops every day by cheat (as it's quicker than having Dahlia do it - So little time!). And if they aren't dirty enough to "Make Clean," I temporarily use "Make Dirty" on them while the game is paused. Then I "Make [them] Clean" again. I've done that since I bought Cottage Living.
Also, I have found that if you have a sprinkler nearby, that will help them get clean also.
Thank you! I didn't know that about the sprinkler, either. That's very helpful.
While I attempt to work out possible causes of the problem I stated above, I'll add a couple of sprinklers to the (flat) roof, where I house my chickens. Hopefully, there won't be many puddles forming up there while Dahlia (my sim) is at work.
Clean the coop. When they go in, they'll be clean again.
Thank you, too!
Do you know whether Shift-clicking on the Coop and selecting "Make Clean" (with testing cheats enabled, of course) will count? Or do I have to have my Sim do it herself?
