In addition to all the suggestions, you may also consider tying up a gutter along the bottom edge of the tarp onto the fence (not sure how far down the tarp is now in winter). You can divert the water coming down from the tarp away to the other end of of the run.
The concrete blocks along the bottom edge of the fence (like
@3KillerBs suggested) will also help.
I had gone through the frustrating muddy run problem in our chicken run last year, trying all kinds of bedding, in the end, it's the makeshift overhang and gutter that solved the problem! The run has been completely dry since.
Attached is a photo of the overhang on top of the cube.
The tarp is inside the cube (first time chicken parents, didn't know the run should best be covered! so tarp covering was an afterthought). I use an internal gutter to catch the water that comes off the tarp and divert it out of the cube.
Sorry about the sad looking squash. The photo was taken a day after the first frost in the fall. We cut a 4x8 plastic coroplast sheet ($30 in homedepot now, it was $20 then!) into two 2x8 strips. So my overhang was a $40 solution!
Gutter is < $6 for 10 ft.