Time-resolved hard X-ray photoelectron spectroscopy (trHAXPES) using microfocused X-ray free-electron laser (XFEL, hν = 8 keV) pulses as a probe and infrared laser pulses (hν = 1.55 eV) as a pump is employed to determine intrinsic charge-carrier recombination dynamics in La:SrTiO3. By means of a combination of experiments and numerical N-body simulations, we first develop a simple approach to characterize and decrease XFEL-induced vacuum space-charge effects, which otherwise pose a serious limitation to spectroscopy experiments. We then show that, using an analytical mean-field model, vacuum space-charge effects can be counteracted by pump laser-induced photoholes at high excitation densities. This provides us a method to separate vacuum space-charge effects from the intrinsic charge-carrier recombination dynamics in the time domain. Our trHAXPES results thus open a route to studies of intrinsic charge-carrier dynamics on picosecond time scales with lateral spatial resolution on the micrometer scale.