Had the entire front yard sodded last season. I had asked the landscaping company to remove several inches of the existing soil and replace with clean soil before laying the sod as I knew that there was a lot of clay. They didn't (I fought them on this and got a substantial amount of cash knocked off the bill). As you move from the foundation toward the street, the sod gets worse and worse as the soil cannot drain. A tree in the parkway that shades the ground closer to the street doesn't help either. I considered replacing 1/3 - 1/2 of the sod (after abating some of the clay) this summer but someone told me that the application of gypsum may help to break down the clay and improve things.
Well, now, there's what's right and what's right and never the twain shall meet.
If you decide to remove the sod (you can rent a sod cutter and just cut and place it to the side and then reuse this sod), then I would recommend you put down a 3-4 inch layer of compost or peat moss, along with maybe 1 inch of sand. This combination will then need to be tilled into the existing clay. Then I would regrade and once the sod is put back, you will need to use a roller and roll it flat. Water at least twice a day during dry months until roots take back in.