We have mosquito and drainage issues at my place too. Here's a distillation of everything I have learned.
*Mud and mosquito control always starts with prevention. Sadly, you can't control the weather. Consider ways to raise your run level. Improve your drainage by channeling water away from your coop and run, perhaps digging French drain and/or berms. Build a drainage base into your run, filled with gravel and sand covered with bedding that dries quickly.
*Turn out all standing water daily, including waterers. Cover all rain collection barrels with mosquito proof netting/screens.
*Use mosquito dunks (safe for wildlife and livestock) in pools and ponds.
*Use a Dynatrap or similar device. The bugs caught in it make a high protein snack for your birds!
*Encourage insectivorous bats and hummingbirds to frequent your property with appropriate housing and food sources. Bats can eat their own weight in mosquitoes daily. And yes, hummers raising chicks eat bugs, especially flies and mosquitoes snatched midair!
*Plants that are purported to repel mosquitoes aren't often really effective for me, but worth a try? Lemon, menthol, eucalyptus and mint scents are most often recommended.
*Keep grass and brush trimmed. Tall grass and weeds harbor the most mosquitoes "on the hunt".
*Wear DEET insect repellent. Reapply often. I like the wipes as they offer better control and go further with less waste. Keep them in a freezer ziplock bag since the containers they usually come in allow them to dry out too fast. If they dry out, pour a bit from a pump spray bottle in to refresh them. You can reuse each wipe several times before tossing.
*Yellow lights don't attract as many bugs as white or blue lights. Strategic placement of lights helps.
*Area pesticide foggers should be a last resort. Follow all label directions and precautions very carefully.
Mosquitoes are probably the most dangerous animals on the planet, including man! They have killed more and destroyed more throughout history than anything else. Right now, North America is on track for a very bad situation with mosquitoe born illnesses.
I hope you can get a handle on your challenges. If I can be of any more help, I'm happy to try.