Treatment would depend on identifying the cause, there are several fungal diseases that can affect dogwood, and also non-fungal problems. That's where taking samples to the Extension office would be helpful. Can you think of anything that is different between the parts that are affected and not affected? Does part of the plant get overhead watered by a sprinkler system and the other not? Are any herbicides used on an adjacent lawn? Does the struggling part get dripped on from the roof? Have you tried to scratch off any of the bumps to see if they're scale insects (they don't look quite symmetrical enough to me but hard to tell from a photograph).
Edited to add, is there any hint of colour on the white bumps, kind of a coral colour?
