Hi Woodruffsdad and welcome to the Forum.
If you have a clay soil and no roof to your run, anything you can do to the base may give some temporary relief from the mud but will not be a lasting solution unless you installed a complete underfloor drainage system. This would be a lot of work and probably uneconomic compared with the cost of roofing the run with clear plastic corrugated sheets. Also it wouldn't solve the problem of wet and miserable chickens. I had my run roofed with corrugated sheets last year when it was so horribly wet, and have been very grateful for the dry and comfortable conditions it has provided. Its good to fit a gutter along the lower side of the roofing, perhaps draining into a water butt - ours has a hose attached which then takes water straight into our pond. A gutter helps remove water before it can blow in under the edge of the roofing and on to the floor of the run.
For additional shelter from wind and rain I recommend fixing clear plastic Monotex tarpaulin to some of the sides of the run , which will prevent rain, snow and wind blowing in - I have left one side of my run open so the girls can see out, but the rest is now like a sort of 3-sided tent, very comfortable and the floor stays blissfully dry. Its a lot better for the girls, and now I really enjoy going in there in all weathers to see them, without rain dripping down my neck. You can get the tarp from Tarpaulins Direct see
http://www.tarpaulinsdirect.co.uk/tarpaulins/tarpaulins/monotex-tarpaulin - green is illustrated but they also do clear.
, it's very cheap but easy to cut and fix to woodwork with battens of wood off cuts, and can be removed in summer when it gets hot. I've had the same stuff in use for three years now and its as good as ever, very strong stuff. Some people create tailor-made panels from corrugated plastic on a frame, which can be fixed in place and taken down in summer, but of course you need more room to store these than a folded tarpaulin.
I suppose the Flytes of Fancy shelter you linked would give some protection if your birds were out on a large open area, but I would have thought that, for that price, you could have bought a cheap shed, or bought the materials to make something larger and just as effective. Or maybe the money would go towards buying your corrugated sheeting to solve the problem altogether?
You could also cover the floor with permeable landscaping fabric - search the web for the wide sort, mine is 2-3 metres so fewer joins. This needs pegging down well at the sides to prevent the hens from digging up the edges, and you need to fold under the cut edges to prevent fraying (or pre- sew them to size if you have a sewing machine.) The membrane lasts for years if its covered with a good depth of bedding, mine has now been down for four years and is still working fine. This will cure the mud problem as the earth itself will be under the membrane and the chickens won't be able to scratch it into whatever you put on the floor. You can them keep it clean by daily poo picking, without mud being involved. You could of course do this to your run as it stands, without roofing it, and it would help with the mud, but you might get puddles forming on the surface depending on how much rain there was and how bad the drainage. If a roof could go on to the run, the mud underneath would dry out in time and you could add Aubiose or a similar product, or wood chippings, to the floor. Membrane by itself wouldn't work so well, it would need a 4ins depth of wood chippings or Aubiose or similar on top, but this would be a lot better than mud. If roofed as well, it would then be warm, soft and comfortable for the chickens, whatever the weather outside, and they would be able to scratch in it and enjoy just sitting round in a sheltered corner after lunch, out of the wind and rain. Mine do this and have long quiet conversations, I really like watching them.
09
