Answer: Nectarines and peaches will "bleed" a gummy sap from the fruit whenever it is injured. The most likely culprits are stink bugs (and their cousins in crime the leaf-footed bug) or the plum curculio (a dark brown to black weevil about 1/4 to 3/8" long).
One way to check for these pests is to go out early in the morning, spread a white sheet beneath the tree and give the limbs a sharp jarring strike. This will knock the bugs out of the tree and they often will lay there almost motionless in the cool of the morning, allowing you time to get a good look.
Another possibility is that the substance is "honeydew", a sticky substance secreted by aphids and scale insects. Examine new leaves and twigs for signs of insect pests.
Depending on what you find, a recommendation for control can be made. Feel free to submit another question, including this original information plus added observations, to the website.
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