Perhaps gummosis, common in prunus species. And/or overly wet soil.

We did have several strong downpours this last week, but the tree is in a pot because of the poor drainage in the yard.

Looking up gummosis, it seems that there is no direct treatment. I will scrape away at a spot of the ooze and look at the underbark for signs of insect or death from fungus. It is oozing at several spots on the main trunk -- I fear that I will lose the tree. And this is the first year it has set fruit!

Perhaps I should just take it out so it doesn't spread to the peach, although that is about 50 feet away.

Some borers can do this...not sure if you're effected, here we see this from winter injuries.

The reason I quit fruit trees. The very second the tree is weakened from anything from stress caused by MN, or not quite for your zone, or critters living in or trespassing in your yard or some kind of disease. The borers will attack. If you don't do your scheduled spaying, then once they get inside, they are pretty much at home.

Only the methley plum out performed the little suckers. They still attack it, but the tree grew so fast, it didn't really matter. My peaches, apples, nectarines all were attacked at some point and the borers tore them up to a point I just cut all of them down. The problem was I didn't live there.

Good Luck.