B. Rain running off the roof of the coop/other building into the run?
C. Water running along the ground into/through the run?
D. Water accumulating in a low spot on the ground?
The solutions for all these are related, but different.
First, fix drainage problems -- install gutters, use grass swales or French drains to divert groundwater, and grade the area to ensure that water drains away from the run instead of into it.
Second, if you live in a climate that gets a lot of rain, either all year or seasonally (you can add your general location to your profile to get more targeted advice ), you might want to roof part or all of your run.
Third, if your coop is in a low area that cannot be drained with the measures suggested in the first step you should consider either moving your run out of the low-lying area or building up the ground by some means.
Fourth, add plenty of absorbent but well-drained bedding. Aged wood chips are the ideal foundation for a deep litter system (many threads are available with useful information). Here in the North Carolina sandhills I have a near-limitless supply of pine straw, which makes excellent bedding in our wet winters because it drains freely and the top dries quickly.
Using a mix of materials with different sizes and textures -- wood chips, wood shavings, straw, pine straw, fall leaves, etc. -- reduces the chance that the bedding will pack or mat and thus create stinky anaerobic pockets.
Finally, if you're doing everything right -- good ground drainage, plenty of bedding, etc. -- and the run is simply being overwhelmed by extremely heavy rain, you can buy a few bales of straw or hay and set them in the run in various places for the chickens to sit on up out of the mud.
If you've got their feed and water in the run you could set them up on a pallet so they can eat and drink out of the mud.
A few concrete blocks or pavers can solve spot mud problems such as the hole that inevitably forms at the base of the run but will have to be lifted and replaced every time you add more bedding so they don't become buried.