It appears that your plant is under attack by a host specific chewing insect known as the Bougainvillea Looper. This a common pest on bougainvillea in your area. Pick off and dispose of the little critters as you spot them on the plant. One option would be to treat your plant with Bayer Advanced 3-in-1 insect, disease, and mite control. Some organic options include controlling a caterpillar problem with Bacillus thuringiensis (Bt). It is a bacteria that basically eats the caterpillar from the inside out and works as a contact spray or spray with Spinosad with a 7-10 day residual that should be sprayed in the late afternoon/early evening after the bees have returned to their hives. Once the spray has dried, it is safe for beneficials.
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Bayer Advance products should never be suggested because of their long-term residual toxicity affecting bees and other pollinators, including hummingbirds. This product makes plants toxic for several YEARS following one application.