Don't put sand onto mud. It will disappear in short order leaving no trace of the expense or work you put into it.
Put sand (or gravel, or a sand/gravel mix) on bone DRY ground for a good long-term muddy run fix.
For now, while things are real wet, the best first-aid is some sort of coarse organic material. Ideally coarse wood chippings or coarse mulch, but something finer like tree chippings or straw can also work as long as you rake it out and replace it before it starts to get too nasty.
Also do everything you can to limit water input. Install gutters (yes, even on a small coop
) and downspouts to direct roof water well away; put a cover on the run if possible; and trench around it (ideally leading the water away to lower ground).
Pat, in a mud swamp herself and expecting Barn Flooding Problems tonight with anohter 1" of rain, sigh
If you are looking for a way to walk through without getting muddy, you could put in rocks, cinder blocks, such as that, as stepping stones. They will eventually disappear into the mud, especially as you have a freeze/thaw cycle, but could be a temporary help. I have plenty of free rocks if you want to come to Arkansas.
You could build the level of the soil in the run up to help it drain, but they will dust bath in it and create puddles so you will constantly be filling mud holes. And putting loose soil in right now would really make it muddy. A dry season project, then cover it with sand.
I really don't recommend hay or straw! I tried that last year and the chickens just mushed it down into the mud. The muddy top dried, but the hay (about 2 inches down) stayed wet and soured. It was a stinking mess that I had to dig out a month later. And in addition to the soured smell, it had fungus and stuff growing in it... not something you want chickens to eat.
Wood chips, on the other hand, seem to work better. In a pinch, that's what I use.