A recent, major, puzzle in the core-level photoemission spectra of doped manganites is the observation of a 1 - 2 eV wide shoulder with intensity varying with temperature T as the square of the magnetization over a T scale of order 200 K, an order of magnitude less than electronic energies. This is addressed and resolved here, by extending a recently proposed two-fluid polaron - mobile electron model for these systems to include core-hole effects. The position of the shoulder is found to be determined by Coulomb and Jahn-Teller energies, while its spectral weight is determined by the mobile electron energetics which is strongly T and doping dependent, due to annealed disorder scattering from the polarons and the t(2g) core spins. Our theory accounts quantitatively for the observed T dependence of the difference spectra, and furthermore, explains the observed correspondence between spectral changes due to increasing doping and decreasing T. Copyright (c) EPLA, 2008.