I didn't vote because my first reaction is a conditional "yes" -- except that the conditions make a great deal of difference.
Yes, because I think that, in general and for most situations, a covered run that is partially solid roof and partially wire is the best overall option for most chicken-keeping situations because it allows the chickens to make their own decisions about how much shelter they want at a particular time (chicken know how to be chickens better than we know how to be chickens).
BUT, before I'd say anything more definite I'd like to know more about your specific situation.
First, where, in general, are you? Climate matters tremendously -- especially when talking about a mud problem.
Second, what is your soil type? Sand, loam, clay? Deep, shallow, gravely, rocky?
Third, what is the drainage pattern for your coop and run area? If it gets muddy
every time it rains I have to suspect that you either have the chickens located in a low spot where water collects or have a drainage pattern running through the chicken facility instead of channeled around it.
Unfortunately, neither roofing the run nor changing the bedding can fix a mud problem that is the result of the land's drainage issues. BUT often those drainage issues can be fixed. If you show us photos of your coop and of how it sits in the surrounding land we can help.
Fourth, while any dry organic material can be used as bedding in the coop and litter in the run, hay would be very low on my list of choices for use in a run that ever gets wet at all. It's not very absorbent compared to other options and breaks down faster than almost any other possible material.
For many people, the gold standard of litter for your run, especially a run that ever gets muddy, is coarse wood chips -- the kind you get from a tree trimming service. They are much more absorbent than hay or straw and break down slowly.
Additionally, from a composting perspective, good-quality hay (nice and green and full of nutritional value for the animals who eat it), is a compost GREEN -- a high-nitrogen element. Manure is also a green. What you need for your run litter is a compost BROWN -- a high-carbon element.
When you combine chicken manure, the green element, with wood chips, straw, pine straw, dry leaves, or other such browns they react together to digest the manure -- eliminated odor and creating garden gold.
Switching bedding won't solve a serious drainage problem, but since you mentioned an odor problem I suggest that you try swapping out the hay for a high-carbon bedding as a first step.
