How do you remove cloudy white patches from a polyurethane finish?

Try using a small amount of Rubbing Alcohol. Brush or dab a small amount over the white marks, this may help release the moisture by softening the finish. Always do a test area first.
If the product has cured and will not release the moisture try acetone or nail polish remover in very small amounts to affected area only.
If these don't work get out the Furniture Stripper and start again.
Try and avoid polyurethane!
Try this product or something similar: Varathane Diamond Interior Finish, Water-Based