Molecules can sequentially absorb multiple photons when irradiated by an intense X-ray pulse from a free-electron laser. If the time delay between two photoabsorption events can be determined, this enables pump-probe experiments with a single X-ray pulse, where the absorption of the first photon induces electronic and nuclear dynamics that are probed by the absorption of the second photon. Here we show a realization of such a single-pulse X-ray pump-probe scheme on N$$_2$$2molecules, using the X-ray induced dissociation process as an internal clock that is read out via coincident detection of photoelectrons and fragment ions. By coincidence analysis of the kinetic energies of the ionic fragments and photoelectrons, the transition from a bound molecular dication to two isolated atomic ions is observed through the energy shift of the inner-shell electrons. Via ab-initio simulations, we are able to map characteristic features in the kinetic energy release and photoelectron spectrum to specific delay times between photoabsorptions. In contrast to previous studies where nuclear motions were typically revealed by measuring ion kinetics, our work shows that inner-shell photoelectron energies can also be sensitive probes of nuclear dynamics, which adds one more dimension to the study of light-matter interactions with X-ray pulses.