If you can fill in the low spots over time, that should help the water in the run drain away. It's just hard to do that if it's pouring rain at the moment. I have sand on the surface of the bare area in my run next to the coop and like it.
When I first moved here, I had some troubles with the yard. The first spring when I had areas that were flooded, I dug a tiny channel, only the width of my finger, to let pooled up water drain away to a lower area. I live on the side of a hill, so if I went far enough, that worked. If there's an area close to the run that's lower, you might be able to do something like that, just by jamming a flat shovel into the ground and wiggling it back and forth, in a line to the lower area, where it might drain away.
If your run is just in a low area with higher ground all around it, then you're stuck building it up, so it's higher. I ended up regrading a few spots, too. Giving the chickens pallets to stand on can make them more comfortable temporarily, until you can get something worked out.