Yuck! Don't know what it is, but spray with Neem Oil immediately.

The disease seems to be Powdery mildew. It is caused by the Oidium fungus. It appears when there are prolonged periods of cool, dry temperatures. The symptoms of the disease can be identified with the appearance of a white, powdery-like substance on the panicles, new fruit and the undersides of new leaves. This disease can cause premature leaf and fruit drop. Mature leaves that are infected have spots that appear a purplish-brown color. This occurs as the white fungus begins to disappear.

The treatment for this fungus is a fungicide program that begins in early spring and extends to the end of the crop season.

Thanks for the replies! Powdery mildew makes sense, so I'll try using neem oil consistently (maybe with sodium bicarbonate, too, since I saw that suggested elsewhere).

Just wanted to show an updated image of my mango. It's been looking much healthier in recent weeks, and it just exploded in new growth!