Not sure how muddy your run is or if you are needing to fix it before the upcoming rain but........... I would take some muck boots in there and start by turning the dirt er mud over to loosen it and help it to drain some. I would also look at straw or hay to put on top of it so they have something cleaner to walk on. You could also perhaps if the water is deep in areas or it is just really soup like ......... take some pallets and to protect the feet of the birds screw some plywood to the top to lay in the run so they have a place to get out of the mud.
I turn the dirt in my run frequently to keep the smell away and they love digging in the freshly turned dirt.
I think the straw/hay would work well for the geese. (Leaning toward grass hay so it is softer on the feet)
Good ideas! I will try them. I let the birds free range during the day, and they are so fast to get out of the swamp. Poor birds.