The general intermediate state representation (ISR) for single-electron ionization is adapted to the case of K-shell (or core-level) ionization in molecules. The development is based on the so-called core-valence separation (CVS) approximation leading to a considerable simplification of the ISR secular equations. Using the CVS approximation the core-level ISR can be formulated entirely in terms of the intermediate states of the valence electron excitation problem, which allows one to construct consistent nth-order approximation schemes for the (single-hole) ionization energies by a specific extension of the (n-2)-nd order ISR approximation for electronic excitation. In particular, the CVS-ISR concept is used to derive a consistent fourth-order approximation for core-level ionization based on the existing second-order algebraic-diagrammatic construction [ADC(2)] approximation to electron excitation. The computational scheme combines the diagonalization of a Hermitian secular matrix with finite perturbation expansions for the secular matrix elements. The explicit configuration space is spanned by one-hole (1h), two-hole-one-particle (2h-1p), and (3h-2p) ionic states with exactly one hole in the core-level shell of interest, while the configurations considered implicitly via perturbation theory extend to the class of 5h-4p states. A characteristic of the method is that the dominant valence electron relaxation effect is accounted for at the post-Hartree-Fock (HF) level. This calls for the relatively high order of perturbation-theoretical consistency, but avoids, on the other hand, the necessity of a localized (symmetry breaking) one-particle representation in the case of molecules with equivalent 1s orbitals. The method is size consistent and thus suitable for applications to large systems.