Clay holds water, obviously, but water does slowly seep through it. Clay makes good lining for ponds - you have four of them - they probably vary in depth from time to time.
One expensive yet effective solution is to hire an excavator to dig one large pond and use the removed clay to fill in the other three low areas. Since the one big pond is going to be much deeper than any low area you have now it should hold a large enough amount of water for fish (perhaps trout) and ducks. Ducks look good around horses.
Since this is a solution you have probably already thought of, it may not be the one you are looking for.
Compacted clay also makes a decent bed for gravel, perhaps you could line the perimeter of the new pond with gravel so the horses have good footing when they come to drink.
Although I am a mechanical engineer, I grew up on a small farm with horses, ducks, etc. and ponds are a very nice feature to have on small farms.
You may get lucky and discover gravel beneath the clay if you dig deep enough which you can use to line the perimeter. And if you are really lucky you may discover a spring in the gravel which would ensure fresh water for your horses.
Good Luck.