not sure from the pics of the size of the run , but the easiest way would be too lay paving slabs . You just need to clear the muck off and lay the slabs as and when you get them . No need to do it all in one go , just as you can either buy them or get them off of something like freecycle .
Broken ones will do , just lay them like crazy paving . All you need do is clear the area you are going to lay the slabs on , then chuck down a load of sand and roughly level it , and then lay the slabs ...job done .
There after you only need sweep with a stiff broom or even hose or pressure washer it down when needed. You may need either a drain off area or a sump to take excess water , both easy to do over time .
I know it isn't ideal , and doesn't look nice and natural , but to look natural you need a huge area with only a few birds on it .
I have seen people use gravel etc , but it bloody stinks like hell after a time . Slabs are much easier to live with , and not hard to lay either . Some builders yards sell broken ones off really cheap (some don't ! ).
I laid my shed floor and an area outside , with flat stones from the river that runs through my land ...works a treat and was free !!!!
Hope it gives you some ideas Annie ...
cheers
Russ