Quagmire run

With the prolonged rain, the run in the garden about 3m x 4m has become a quagmire. It has been there about 18 months and was quickly denuded of grass and I have put bark and wood chips on the surface to create a more "crumbly" texture good for scratching in. It is now mainly a slippery mud arena particularly near the door where the girls wait for food to arrive and to be allowed out to free range. I have been putting down small branches to provide a slip reduced surface for me to enter to feed, clean and collect eggs.
When this rain ends should I dig the surface mud (& chicken poo) out, dig in sand or gravel to make drainage easier, add more bark/wood chips or what else can be suggested. I have been reading with interest "and so it continues to rain"and I may have to consider rain proof covering (only have netting to keep the greedy pigeons out at present). There is a complete covered coop inside - the sort with an upstairs nesting box, underneath run and roof - which I considered too small for 3 hens to enjoy on its own. My husband was fed up of them free-ranging into the kitchen so he built Chickatraz the 3x4 run but I still let them out with orange workmens fencing net to keep them to the top end of the garden.
With the mud, and if they lay eggs during the day the eggs get soiled with muddy feet. I wipe them off with a damp cloth but it has occurred to me there will be chicken poo in the mud. Is it still safe to store and use these eggs?
Also the hens are looking rather dirty (especially the white one), they get wet and muddy pushing under trees and shrubs. Can you bath hens or should I leave them to clean themselves or just leave them, as they don't appear to be worried about their appearance.
Thanks for any advice 
Jaws

I had my girls in a covered run which helps matters no end. I could cover the sides, front and back, as necessary with those green tarps with eyelets. For the front I invested in a clear tarp which served as their window on the world but kept them rain and draught free.  I always had one side of the run without protection to ensure fresh air and plenty of ventilation.  Straw bales in one corner provided something for them to stand on and to go under if they wanted extra warmth.  On the floor I used Aubiose of Hemcore which does help keep muddy feet out of the coop and helps keep eggs clean. 

Your hens don't mind getting wet. They don't like being cold and wet though.  The eggs will be fine.  I cleaned any dirty ones off with clean straw

Don't bathe your hens until the weather really warms up.  If you provide them with shelter they can then dry themselves off.  Provide them with a dust bath too to help them clean themselves

Mine is pretty much the same - we have one side of the run uncovered and the darn rain comes at the only angle that it can get into the run!

I keep turning over the ground and was getting really worried about the dirty hens, but today we had a dry day and it has dried out a lot.  The hens seem to come out of the coop clean every morning even though they go in all muddy at night!

My hens seem to prefer being out in the wind and rain, they look a sorry lot all wet and bedraggled but there are plenty of dry places to shelter. My goose looks like he's been bedazzled he keeps laying under the cherry tree and the blown down pink blossom sticks to the front of him.