The thick mass-shell model with a two-dimensional singular inner surface introduced by de Felice and Yu (1993) to describe a static incompressible fluid arbitrarily close to a black-hole degeneracy, is extended here to sources with a polytropic equation of state. We find that the properties of the mass-shell solutions change considerably from small to large polytropic indices. The definition of gravitational energy given by Lynden-Bell and Katz (1985) is found to be applicable to our solutions and, in particular, we show that the remarkable implication of that definition, namely that `all the energy remains outside the black hole' is consistent with the behaviour of the matter and field energies inside our shell configurations as they shrink to the black-hole limit.