My fields are on good old Essex clay. It does get very wet and sticky and deep in places in the winter, and bakes bone hard in the summer.
I have enough land to spread the poaching, but the grass does disappear in certain places this time of the year.
We wait until it dries out a bit more towards Spring, then my husband harrows it to knock the rough ridges flat. We only ever roll the very worst bits, because if you continually roll the ground, then it compacts down even more and this makes the drainage even worse, so you shouldn't do this on clay type soil.
I sweep my hay seeds up when the horses have finished their hay in the shelters, and also from my hay shed, then I spread these seeds out on the bare bits - don't know if it does any good, but certainly doesn't hurt the field.
If your field is very bad, you may have to fence some off and re-seed. The problem is, although the grass does grow back, so do the weeds with a vengeance, especially if there isn't a good thick grass (sward) to help keep the weeds away. You may notice that grass continues to grow all through winter, depending on the temperature. I think grass stops growing when the temperature is below 7 degrees???
Hope this helps.