Not tarpaulin for the roof, except as an emergency temporary measure, as when it rains it forms huge, heavy puddles between whatever cross members you fix across to support it. I tried this over my mesh roof the first year I had my new run and it deformed the mesh so it sagged down into heavy depressions full of many litres of water.
Wooden boards would be strong and would give shade, but unless you already had some to recycle they would probably be overkill as far as waterproofing was concerned, and also very heavy compared with alternatives- could the framework support them? Also possible redmite?
Most people use either corrugated opaque sheets (sold as Onduline, Or Wickes do a nice own-brand one in attractive green) or corrugated clear plastic sheets. The onduline type need more support as they are heavier and will bend out of shape if not given enough crossbeams at the recommended distances, but are long lasting, give shade to the run, and of course solve the water problem. But they will make the run darker, being solid. The clear plastic kind are lighter and easy to fix, let the light in, which is especially good in winter. They do make the run hotter on sunny days. This is a bonus in winter, of course, but in summer, if we get any, you need to provide alternative shade for the hens. Whatever you use needs guttering as well, perhaps to a water butt or a leaky hose to nearby garden beds (mine is fixed up to drain into the pond.) I didn't fix a gutter at first and found the rain just blew in the side of the run off the bottom edge of the roof.
This summer I've just succeeded in getting my whole run covered (it's about 4 X3 metres) and am already noticing a fantastic difference in the comfort of being in there, for me as well as the girls. Everything is always dry and comfortable, and now I can use Aubiose, which is shredded hemp stems, very soft and absorbent, on the floor. The girls love excavating it into huge heaps and so far, although I poo pick daily, it seems to stay lovely and clean and I think a nice deep layer will last a long time, but now i could have the choice of any sort of litter, without having to think about the effect of rain on it. So if you can manage a roof, whatever kind you choose, I'd say go for it before the winter sets in. If you cant manage the whole lot, it would be great to do at least one end, and perhaps cover the sides there with plastic tarpaulin, to make a dry sheltered end for the dust bath and feeder, and a perch there for the hens to hang out on in bad weather.