Well hay or straw dose not do well to soak up urine. Ever muck stalls that have straw as bedding. Get towards the bottom and you find straw dripping with urine. ( depending how dirty the stall are I guess) straw also does nothing for the smell.
It doesn't have any absorbency power, and does nothing for keeping smell down.
Adding it to bedding is fine to give them nesting material though. I use shredded paper because it's free. 50lbs of hay from Tractor supply is only $5, but free is better.