Do you have photos of their feet?
Do they free roam, live in a coop/run, etc.? Photos of your setup may help too.
A lot depends on your setup, straw may be beneficial to spread across muddy areas, but having good drainage will go a long way in keeping birds out of the mud too. Having a portion of run covered so there's dry areas can be of benefit so birds have a place to get onto dry soil/bedding too.
As for cleaning feet, unless they have large poop/mud balls attached to the nails/toes, I would not wash feet in those temperatures.
If nails need trimmed, you can just wipe off excess dirt, then trim with clippers, taking care not to trim too much getting into the quick. I've found if I have a hen with nails that have overgrown a little it's best to trim, then repeat in a few days until I get the nails back in shape.
https://www.backyardchickens.com/articles/beak-and-claw-trimming.74570/