  For the calculation of core-excited states probed through X-ray absorption
spectroscopy, the core-valence separation (CVS) scheme has become a vital tool.
This approach allows to target such states with high specificity, albeit
introducing an error. We report the implementation of a post-processing step
for CVS excitations obtained within the algebraic-diagrammatic construction
scheme for the polarisation propagator (ADC), which removes this error. Based
on this we provide a detailed analysis of the CVS scheme, identifying its
accuracy to be dominated by an error balance between two neglected couplings,
one between core and valence single excitations and one between single and
double core excitations. The selection of the basis set is shown to be vital
for a proper description of both couplings, with tight polarising functions
being necessary for a good balance of errors. The CVS error is confirmed to be
stable across multiple systems, with an element-specific spread for $K$-edge
spectrum calculations of about $\pm$0.02 eV. A systematic lowering of the CVS
error by 0.02-0.03 eV is noted when considering excitations to extremely
diffuse states, emulating ionisation.
