  We consider a variant of Gamow's liquid drop model, with a general repulsive
Riesz kernel and a long-range attractive background potential with weight $Z$.
The addition of the background potential acts as a regularization for the
liquid drop model in that it restores the existence of minimizers for arbitrary
mass. We consider the regime of small $Z$ and characterize the structure of
minimizers in the limit $Z\to 0$ by means of a sharp asymptotic expansion of
the energy. In the process of studying this limit we characterize all
minimizing sequences for the Gamow model in terms of "generalized minimizers".
