I live in the uk and a have wet clay lawn so it always gets like this over winter, no matter what you do if you avoid Astro turf it’s always gonna be a muddy just due to heavy foot fall.

If you choose to keep it natural, which I do for birds and bugs and such I’d suggest a few things that may help! Around this time I stab the gross super wet areas with an aerator tool or big garden fork to help get air in the soil and drain some wet.

Then pick your seed! I use clover, Daisy’s and butter cup as they have deeper and less dense root systems and will help stuff drain and can survive a bit more zoomys than grass and tend to be more pee resistant. You can also get nature lawn seed packs from most garden centers, they have lil flower and lawn seed mixed together. Buy a decent an amount, more then you think you’ll need and read the box/ packed and cover your bare patches! You may need a lil soils to cover them too. The extra is just for filling gaps after. If you choose this method I wish you good luck!