I honestly do not know. Off the top of my head I would look into "Peach Leaf Curl" but I don't have my plant disease handbook, that stayed with the farm when I sold the farm. Look at the underside of those leaves, where the spots are, are there insect bites at those points?
Can you see any insects on the leaves or crawling on the stems?
yes to the above, you have an insect, use a lens to magnify enough, see if you can identify. Otherwise a general purpose spray will usually work. For instance a Pyrethrum type spray.
Do the leaves feel "dusty"? Often indicates mites, spider mites, flat mites, two spot mites, and other sap sucking mites. These are technically arachnids, rather than insects, so an "insecticide" won't work, need a "pesticide" that lists mites on the label in order to kill mites. Mites are small enough I have trouble seeing them with my eyes, but I recognize the feel of a medium to heavy infestation, they give a dusty texture, especially to the undersides of leaves.
Actually, I do suspect thrips, they are quite small, Wikipedia has a page on them. They are more common as an indoor plant problem, but can be an issue outdoors or in greenhouses.
treat for thrips, only if that does not work try the treatments for the fungal infection peach leaf curl. Peach leaf curl only occurs in peaches, nectarines, almonds and apricots. Ume is most closely related to apricot, even though it is sometimes called a flowering plum. Peach leaf curl does not infect plums at all.
en.wikipedia.org
en.wikipedia.org