This may sound too simple, but it does work with heavy water spots, calcium build up, and crud (like oxidation), shower walls, and some acid rain gunk on automobiles ... get a soft cloth or a sponge and simply put liquid Dishwasher liquid on it and rub it (nice and wet) over the are you want to clean. Let it sit a few minutes and then simply rinse and wipe off. Liquid Dishwasher soap is made to cut through stuff like that, on dishes, and inside an automatic dishwasher.
You can try a small area and see how it goes. If it does not work, you've not lost anything (especially if you already have a dishwasher at home).
WD-40 works great for oil and tar, Automatic Dishwashing liquid works good for everything else. (And Mr. Clean Magic Eraser works fantastic on your awning)... phew! I think I've just given you enough work!
Good luck!