X-ray absorption (XAS) and x-ray magnetic circular dichroism (XMCD) spectra at the L$_{2,3}$ edges of Mn in (Ge,Mn) compounds have been measured and are compared to the results of first principles calculation. Early textit{ab initio} studies show that the Density Functional Theory (DFT) can very well describe the valence band electronic properties but fails to reproduce a characteristic change of sign in the L$_{3}$ XMCD spectrum of Mn in Ge$_3$Mn$_5$, which is observed in experiments. In this work we demonstrate that this disagreement is partially related to an underestimation of the exchange splitting of Mn 2$p$ core states within the local density approximation. It is shown that the change in sign experimentally observed is reproduced if the exchange splitting is accurately calculated within the Hartree-Fock approximation, while the final states can be still described by the DFT. This approach is further used to calculate the XMCD in different (Ge,Mn) compounds. It demonstrates that the agreement between experimental and theoretical spectra can be improved by combining state of the art calculations for the core and valence states respectively.
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