Start with run off water. Where are the downspouts, do you have gutters on the coop, is the coop in a low spot, is it next to concrete pads?
Providing pictures will answer lots of questions.
What is the source of the water that is making the mud?
If it is just that you have had an extended period of wet weather you can cover all or part of the run and add plenty of dry, organic material as litter to absorb the water. Many people favor coarse wood chips for this purpose.
If water is coming off nearby building roofs into the run you need to install gutters and direct the downspouts away from the coop.
If water is sheeting across the ground into the run and/or there are natural rainwater channels running through the run then you need to fix the drainage -- cutting diversion channels, installing French drains, creating grass swales, etc.
If water collects and pools in the run because it's a low spot in the area then it's not a suitable place to keep chickens without significant re-grading of the land and/or adding enough soil to raise the ground level significantly above the water level.
As a temporary measure, you can put down pallets and intact straw bales for chickens to sit on up out of the mud. You can also use straw bales to create a temporary dam to keep water from flowing across the ground into the run while you are working on fixing the core of the problem.
as others have said, pictures would help. Water in runs comes from three sources. It falls there, it is directed there, or it rises there. Those sources require different solutions, though they can compliment one another.
If its falling there, a roof design of some sort will reduce that, the width of the roof overhang determining how much wind blown rain can make it past the roof. If you have a wall structure (like a shed coop), orienting it on the side of the prevailing summer winds (for most of us) will further minimize that concern - or the winter winds side (for those of us subjected to seasonally heavy windblown snows. Floor covering and grading of the ground determine how wet, and for how long, your bird's feet get during extreme events.
If its being directed there (honestly, this is where the vast majority of problems come from) by roofs and other impermeable structures on the property, ground sloping towards the run or other landscape features that cause water to pool and sit? The you are looking for some combination of grading, berming, trenching to redirect water away from the run, and to encourage it to evacuate the run as quickly as possible when it does enter. Setting your hen house on a "hill", even just a few inches, then grading the ground away from it - just as you would do with your own house - will make a world of difference. In plumbing, a slope of just 1/4" in 12 is adequate to drain waste in your pipes. 3" of fall will help encourage water to find a spot 12' from your hen house, outside most runs. You can then collect it using a trench, swale, or similar to direct it into a holding area until it can percolate thru your ground. Floor covering can help here also, in providing additional space underfoot for water to travel in while keeping your birds out of it.
The final source is uprising. Changes in the water table and temperature can cause water to wick up thru the ground, penetrating concrete foundations and making the soil surface damp. That's stopped by either an impermeable barrier (like the plastic sheet installed under a concrete foundation/floor before its poured) or a capillary break that interrupts the ability of the soil to suck water towards the surface. Leaf litter and wood shavings work extremely well for that, which is why so many of use use them. It also helps with cold composting, helping to mitigate some of the smells of raising poultry. Sand, unfortunately does not - you need only go to the beach to see that the sand is wet at elevation above the tide. Nor does it help with smell, so many use PDZ (zeolite) for that issue. Finally, the ability of sand to drain degrades as the space between the grains is quickly filled with chicken droppings. So I don't recommend it for most situations.
Attach some photos of the run and property generally, hopefully with some elevation references, maybe we can offer more specific help.
I see two basic ways to fix the problem. Stop the water from getting in to start with or get it out once it does get in. Often it is a combination of the two. Without more knowledge of what you have to work with the best we can do is give some generic ideas. They work for some, they don't for others.