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What's good about all this is that it is teaching you responsibility and animal husbandry. . You are experiencing waste management problems now, and what you are learning will be with you the rest of your life
In farming, manure (duck poop) is considered an asset due to the concentration minerals and nutrients. It is often spread out on a field and later plowed under to help replenish the next crop. Other people will compost the manure, to add to their garden later.
If you cover your run with topsoil or straw, you can recover this along the ducks excrement by composting and using it later to provide (sell??) to the neighborhood for their flower and vegetable gardens. This way you turn a waste product into a marketable asset.