why would the substrate be charged? You are probing the TEY or PEY in the material and exciting the core shells using X-rays not electrons? Maybe I do not understand your question. Are you running an in-situ experiment of some sort?
We are doing TEY. We do not only excite electrons from occupied to unnocuupied states, but also some of them are coming out from the sample (inside electrons also provoque Auger and fluorescence and selfabsotption), my questions deals with this extreme: in which ammount can be charged up my sample so as to lose sensitivty. It is not an in-situ experiment, we just put the samples into high vacuum through an step-vacuum transference system.
In principle the question is justified: If you excite the electrons into the vacuum and detect them somewhere else, then of course these electrons are "missing" in the sample, and the sample will get charged. From my experience, this is a dynamic effect. Since the current can be detected (if you perform TEY I suppose you have a gold plate or something comparable), there must be some charge flow through the sample, which will cause slight charging of the sample. Depending on the actual absorption, this current (and thereby the charging) will be larger or smaller, and may thus lead to a slight smearing of the spectral features.
There is an open access paper in Journal of Synchrotron Radiation dealing with this topic by D. Vlachos, A. J. Craven and D. W. McComb (JSR 12 (2005) 224 (doi: 10.1107/S0909049504030146).