Circulation is merely a defined as a velocity integral along a closed path, velocity potential is based on Helmholtz-Decomposition which decomposes the flow field as a scalar and vector potential. The vector potential is basically vorticity. When you ignore the vector potential potential an very useful for estimating the flow field. One step up you can assume the vorticity is lumped into discrete elements and calculate complex phenomenon through Biot Savart law. the connection between circulation and potential is the 2D closed surface integral of vorticity flux is equal to the circulation integral around the boundaries of this closed surface (greens theorem). So usually when you lump vorticity you assign a circulation to which is really a quantity of vorticity. Maybe that answers your questions? Please elaborate....
