but I would try the TSP in a bucket of water and a deck scrub long handled brush.
I would start at the bottom and work up. after an area is cleaned then rinse it thoroughly. As you work up rinse the area just cleaned and push all the stuff down with a water stream off the roof.
Why I do not recommend working from the top down is that the tsp and or bleach will streak the down hill area to be cleaned and you might end up with streaks.
I do not think you will do any good with a pump sprayer. You will need to scrub the areas. Do not use a high pressure water washer.
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