The equivalent-core approximation is implemented in a novel way so as to calculate core-level relaxation energies in photoelectron spectroscopy. The method is based on self-consistent field (SCF) Hartree-Fock molecular-orbital calculations via linear combinations of atomic orbitals, and involves evaluating the difference of sums of two-electron Coulomb and exchange integrals, for all electrons in an atom and in its equivalent-core ion. By thus avoiding SCF calculations with a core hole present (the true final state of photoemission), this procedure is shown to significantly save computing time in comparison with an exact SCF direct-hole calculation. Application of the method in single atoms and selected molecules shows about a 10% difference with respect to direct-hole calculation results. The approximation introduces about 1-6 eV errors compared to the experimental results of gas phase molecules. This method thus should be a generally useful procedure for estimating relaxation energies in core spectra.