So you feel it bumping only when you give it gas, and won't do it when the car is coasting. Sounds to me like it's something in the powertrain.
"Bumping" tells me that there is a problem with your transmission. When did you last have a transmission service? Would probably be a good idea to have it done just to see what happens. If the transmission is shot, then you know. If not, the service won't hurt it any.
See what happens by doing that.