Just going by the 2 pictures, it does appear you have some good slope for drainage. This is just my opinion, so take it for what it's worth. It looks like your biggest problem is water coming in to the paddock from the upper side. In the picture with the white outbuilding, it looks like a natural swale running though the paddock. To me, it would make sense to divert the water coming in from the high side around that area. Whether it would be cutting a swale around, or put in a curtain drain, and let both daylight out to an area, they won't be tromping in, making pucks in the mud to trap water.
Cutting a swale would be the least expensive in the short term. You'd just need to make it deep enough to catch the water, yet, grade it so it's easy to mow. Something 12" -18" deep, then slope back from that, making something like a 5:1 slope so you could mow with about anything, get vehicles close to the building etc. Even drop in a culvert to get vehicles to your hay, and feed storage building. Easy to do with a moldboard plow, and box blade, as long as you have a place to waste the dirt.
Putting a curtain drain across/above the paddock. Dig a trench 18" or so deep, put a 4" perforated pipe in, and fill to the top with gravel. If that is pasture above the paddocks where the horses walk, top it off with 3/4" crushed limestone, or gravel, so it will lock together, and not push around when they walk on it. The water shedding off the uphill side, will drop in that drain/French drain, and divert the water around.
Not sure if the blue building is the one where the stalls are, but if it doesn't have gutters on it, putting them on, and diverting the water away would be a big help too. The smaller white building probably isn't that much of a problem. But, you do need to divert as much surface water away, coming in to the area. Every time they step in the mud, it makes a puck to hold water. And usually with the fine silty dirt associated with a lot like that, plus the compaction from the horses, it seems to be good at holding water, and takes forever to dry out.
Being you're in a different region of the country, it's hard to advise what material to use to top dress the paddock. I'm in Ohio, and Limestone is king here for projects like that. Personally, I use Limestone screenings in my paddock. Basically Limestone chips down to dust. And it makes a big difference which quarry I get it from. A couple quarry's with better screens gets most of the dust out. I prefer to get it from the one's where you'll get approx. 25% dust mixed in. Something that will pack, and lock together. That being said, there are 2 different kinds of Limestone, fresh water, and salt water. Fresh water Limestone will break down quick, and and also turn to mud. Salt water Limestone is harder, and will not break down as fast.
I put it down in my paddock approx. 6" -8" deep. Do it after everything dries up. If you put it on top of the mud, it will just pump the mud to the surface, every time the horses take a step in it. You'll actually want to put the screenings down wet, then roll to compact, once graded out. I see you have a 580CK, and that would make a perfect roller. That's the key to getting 25% or so dust in the mix. If put down wet, it will compact to a very hard surface. Between rolling, and the horses walking on it, if graded to shed the water, it will make a very nice hard surface. I originally did mine 10 years ago, and top dressed with 2"-3" more 4-5 years ago. I clean my padock, mainly around the hay feeder 6+ times a year with my Bobcat, and the bucket skims right across the surface, almost like concrete, almost.... I will skim a little off every time, but it holds up well. Just need to clean it down to the surface, and top dress when it dries up. But, they do need to be put down wet, rolled, and let the horses walk on it for 30 days or so, for more compaction.
I'd move your hay feeder to a higher spot, to prevent them from having to walk through mud to get to it. It appears to be setting close to that swale that's feeding water to the area, only making things worse.
I have 2 pastures, one is the sacrificial pasture I turn them out in over the winter, so they are free to roam. But even in Feb./March during the freeze thaw period, I do shut them out of there when it's extremely muddy. I just keep plenty of hay in the feeder during this time. As long as they have plenty to munch on, they seem content.
Again, not sure what materials are available to you, but the screenings I use don't hurt their feet, and are compacted enough, they won't pick it up in their hooves. With all of that mud, I'm 99.9% sure they're not shod, or your farrier would be there pretty often putting shoes back on. Mud will pull a shoe off PDQ. Probably wouldn't hurt to dress their hooves with some hoof oil. Walking in that much mud/water will wash the natural oil out of their hooves, and cause other problems. Lots of products out there that comes with a brush applicator. I always keep a can on hand, but really haven't had to use it, unless we have an extremely wet year. It doesn't take but 5 minutes a week to apply it, once the hooves are cleaned.
