I have three suggestions for you. First, find one of the recent punch through holes. Using a trowel, scrape away the soil or vegetation around the edge of the hole. Take a few photos. Probe the sides and bottom of the hole. You need to find out if the holes are coming between branches that were left in place, or geologically soft areas. Open the hole by digging with a shovel. Take more photos. Look at more than one hole.
Second, ask for help from a larger, well known contractor. You may have to pay to have them come out. If you get nothing but free estimates, you might get what you paid for. A good contractor will know more about the soils in your area. I was going to suggest getting soil cores performed, but there is a good possibility that the cores could be done between the problem spots, showing nothing.
Third, you might look at installing a track system that would allow the horse to use the perimeter, and keep them off of the center of the pature. This would mean fewer fixes for you. It might be a short term solution, but also means more hay through the warm season. Slow Feeding Horses on Paddock Paradise Tracks - Paddock Paradise Wiki
When grading, I would install drainage swales between the paddock and arena, leading to a swale along the lower side. And another swale between the areana and the pasture. The idea is to take all upslope water and divert it around the next use area. You will need to provide an area for the concentrated water to level out and slow down before it leaves your property - otherwise you may create erosion on the next lower property.