Cleaning a deck with TSP is an option, but I prefer a different approach. Deck brightener products are made to clean decks and remove dark stains, mildew and weathered, grey discolouration. Apply the brightener, then wash it off with a garden hose and scrub brush. Better yet, use a pressure washer. The only issue with the pressure washer is that you need to use it gently. If you have the pressure cranked up to high, or if you place the nozzle tip too close to the deck surface, it can damage the wood by making it fuzzy. Start slowly, with the tip 18″ from the deck, bringing it closer as experience shows you can do this without damage. It’s a finesse thing.
Even with careful pressure washing, fuzzy surface wood fibres may be created. If this bothers you, let the wood dry for a day or two, then give it a quick sanding with an 80-grit disk in a random orbit sander to restore the surface. The wood can be left bare, or finished with various products, depending on the look you want. Washed and sanded wood creates the very best surface for accepting a new finish.