I did have a layer of clay underneath the top layer. So for me it really helped to punch deep holes (Filled with pebbles) in the ground.You might be able to find something like that....but not sure poking holes in the ground puddles will really help drainage, might just make deeper puddles.....unless you were punching thru a layer of clay into sand below.
For 3 years I added lots of autumn leaves, greens and compost on top. And now I have a good top soil where more and more grasses start to grow. This is probably most barley from the grain mix I scatter in the run. Anyway something the chickens don't like very much.
The scratching chickens and rainworms mix the poo into the soil when its humid. In summer we also have glimmering green fly’s who visit the run to eat the poo and I collect the poo if the clay is very dry for a longer period. But in general it‘s a great solution not having to clean the run much. Anyway it doesn't smell, there are no puddles anymore and it looks a lot better as it did 4 years ago. And is still improving.
My run is only covered with netting but there are a few hiding places for rainy weather. E.g an old bench with EPDM on top.