Do NOT use hay to soak up water/mud! The hay will rot, ferment and mold quickly. If you have to use something like that use straw. It will do all the above, but much more slowly and the horses will not be tempted to eat it. (well, MOST horses) You don't want to use it over large areas, just high traffic ones, as eventually, you will have to scrape it up and dispose of it if you put it down thick. A thin layer will help protect grass from cold and damage, but not heaps of it.
I agree with whoever suggested the electric fence. You can use it until either the weather gets better or you finish your fences. It will also save you a lot of money and trouble in the future by moving them around before they totally destroy the grass roots (their hooves go down in the mud and tear it up) If it is torn up enough, the pasture will have to be plowed and re seeded to be good again.