Sounds like mealybugs. And they certainly can harm plants by sucking out volumes of plant sap. I'd suggest that you use one part rubbing alcohol to three parts water and mist your plant with it. You can also remove those insects that you see with a Q-tip dipped in alcohol. The misting will kill the nymphs as well as the adults.

"Dragonflykiller is obviously a crackpot. I will not be cutting down the tree." Lol!! An extremely outdated thread but I couldn't resist commenting on the OP's last thoughts 13 years ago. The though of a greengage plum knocking a cat unconscious I find very amusing :-) Just how flipping big are these plums?? Now if we were talking about monkey puzzle cones, an argument could be made for an injury from falling objects! But a plum?? Seriously??

looks like black aphids