I would mix the TSP according to the directions on the box...if I recall it's a couple of Tablespoons in a gallon of water for general cleaning, and add clorox...I would add a cup of clorox...and that's a strong bleach solution.
Try it...the ingredients are cheap....if it doesn't work, mix another batch using 1/2 as much water.
Linda C

Best home made remedy we use for our decks is a large plastic pail (4 gallon) + 1 gallon bleach + 1 lg box baking soda - wet surface , brush on , re-wet , give light brushing - hose or power wash off - like new at 1/4 the cost of any deck cleaner.
If using store bought - Olympic Deck cleaner is the best we have ever seen and used but at a cost - to each his own