Scrape the spots with your fingernail, then smear the crud on a white 3x5 card (or similar) -- black algae smears dark green.

If it *is* black algae, I'd recommend pressure washing first, and then pouring plain household bleach on all spots. Wait till late evening, and then pour bleach liberally over all spots

Be careful, when pressure washing, to NOT strip the plaster. Old or soft plaster can't take a lot of pressure. Also, try to refill ASAP -- being dry is BAD for plaster, especially old plaster. You're in California -- lots of sun & dry air. You may want to set up a couple of sprinklers or misters on a timer to keep the pool walls damp.