Roofing the entire thing will help, however if you have drainage issues where water from outside the run flows into the run area, you will still have problems with mud or pooling water.
I also use deep litter like many folks above, mainly aged chunky wood chips as a base with dried leaves, dried grass, garden clippings, as I get them. I used to have a massive mud issue and my boots were sinking into it and sticking, and now that's completely fixed and the run stays drier than the rest of the lawn, even though it's unroofed and we get heavy rains. However I have no existing drainage issues in the run, so simply adding the litter naturally fixed it over the course of a few months, as the litter allowed water to drain through into the porous soil beneath, while creating a dry surface on top.
I also use deep litter like many folks above, mainly aged chunky wood chips as a base with dried leaves, dried grass, garden clippings, as I get them. I used to have a massive mud issue and my boots were sinking into it and sticking, and now that's completely fixed and the run stays drier than the rest of the lawn, even though it's unroofed and we get heavy rains. However I have no existing drainage issues in the run, so simply adding the litter naturally fixed it over the course of a few months, as the litter allowed water to drain through into the porous soil beneath, while creating a dry surface on top.