If you're already muddy it may be too late for gravel or sand to be worthwhile, but you can use COARSE organic materials as a temporary measure (like til things dry out next year). Hogfuel, coarse wood or bark chips, that sort of thing -- straw is ok if it's all you can get, but something coarser and thicker would be better. The smaller/thinner the pieces of organic matter, the faster they start to decompose, and really you wanna get it all out of there BEFORE you get much decomposition because otherwise it just ADDS to the mud.
A good thickish layer of whatever you can get (that isn't too pointy on chickens' feet and doesn't contain anything toxic) will keep the birds up off the muck til you can remove it for a more lasting fix. If it starts to get ucky you can aways chuck in another layer on top of the first one.
Also gutters and long downspouts and trenching, if you haven't covered 100% of the bases that way, too.
Good luck,
Pat