Muddy run and fenced area - unsanitary?

[NewToChix]

So our chickens have part of our yard fenced off. They have a coop and fenced in run, but mostly stay in the "yard" which is now just a muddy mess. I have tried gravel/pebbles to try and create a drainage effect but they just scratched it all into the ground. I looked into sand and decided against it for a few reasons. Now, we have planted some sort of alfalfa or wheat grass and it sprouts THICK and heavy which will help with the area. I am concerned that it being muddy/dirt/along with their defecating is creating an unsanitary environment for them. Is this true? Can I do anything to fix it? I shovel poop and hose stuff down when needed, but still feel it isn't enough. Thanks.

Oh and soon we will be adding ducks so that will be a part of the problem as well.