Weeping cherry tree infected
It does appear that you have some dead branches that should be pruned out. It is difficult to tell what the problem, pest, or disease might be. It may be winter injury fro last season, or other environmental stress. any digging in that area in recent months? You have some fungi growing on a dead branch, a sure sign that the branch is dead and should be pruned off. Late season leaf spots are no worry and will not affect the long term health of the tree.
The issue with weeping cherry trees is that they are actually grafted onto a different cherry root stock or base trunk. So, the top of the tree is the portion which flowers nicely and has the weeping branches, while the trunk is more sturdy. I would contact a certified arborist to come and prune the tree, to make sure you don't prune below that graft union. Check online and look for the ISA designation to find someone, as I am not allowed to recommend one over another.
Thank you for contacting Cooperative Extension,