Hi Wallacefan
there are a lot of variables to your question - so the answer is 'it depends'.
If you live in a very hot climate you would want as much shade as possible so depending upon the size of the run, and if your chickens free-range or not -- you may want it all covered. The ground can be left as is. Eventually if your run isn't moveable ---they will graze it down to the dirt if there is grass there. They love scratching out places to take dust baths. If you live where there is a lot of rain, it will be muddy...if you raise chickens with feathered feet and legs -- they will track the mud into your coop.
In my climate, it's impossible to get enough deep dense shade -- except on one side of a building. I have smallish coops that I can move around---but they will eventually use up all the vegetation in the space -- and then moving doesn't provide them with fresh grass. The runs attached to the small coops are quite small---but I let them free range as often as possible.
If your coop has a poop board/tray BTW one of the greatest ideas I have encountered lately is this thread by thespoiledchicken
https://www.backyardchickens.com/t/621363/poop-board-convert-warning-graphic-gross-poop-pictures -- The thread is hugely long, has lots of advice - but the first set of posts - the original poster did such a good job of explaining - that you needn't read the whole thing....unless you have a lot of time. ;O)
If your run is smallish--- you may want a sand covering so you can rake up droppings from time to time. Fully covered will prevent the run from getting wet, will keep predators out like hawks, will provide shade...but size is a factor to consider. IMO you definitely want some parts covered for your chickens' wellbeing.
One last tip - while I am writing - Purina has a feed with omega-3 in-- small pellets instead of the huge ones....and my chickens actually like it-- and they are picky eaters.
Welcome to BYC