Just avoid bogs, areas of the Peak District are notorious. You can spend days getting out of those. If you do get stuck in them, spread your weight out as much as possible and if required use swimming techniques.
Less traffic means it's less likely to be muddy. Bridleways and byways are usually churned up more than footpaths, so picking routes avoiding them is a good idea when things are really bad!
You definitely don't want to ingest all that mud flying around, so try breathing more through your nose to avoid swallowing random detritus.
Let’s face it, it's not avoidable, so you might as well make the most of it! Get stuck in, and cover yourself in trail!