Wait for the mud to dry out as much as possible. Remove any wood piles, dog houses or other item blocking the area from sunlight. Sprinkle absorbent cat litter over the mud and remove it with a shovel.
Install a drainage system if the mud is a constant problem. Bury a perforated drain pipe in the soil at a depth of 12 inches. Lay the pipe so its end comes out in a drainage hole or ditch. Cover the pipe with a 2-inch layer of gravel and fill in the trench with topsoil.
Cover the red clay soil with 4 inches of organic compost. Spread lime over the soil at a rate of 90 lbs. per 1,000 square feet. Do this in the spring before you want to install a lawn so the soil will have a chance to increase the bacteria population essential for plant growth.
Till the compost and lime into the soil at a depth of 12 inches. The compost and tillage will increase the nutrient level and pore space in the soil. The lime will adjust the clay soil's pH. Rake the soil surface so it is even.
Lay down sod over the soil. Select a species that is known to tolerate clay soils, such as fescue or zoysia. Sod will dry up a muddy area more quickly than seeded grass and eliminates the possibility of grass seed being washed away.
Water the sod with 1 inch of water after you install it. Water daily for 2 weeks. That will give the sod a chance to become established.