Element-specific magnetism accessible by synchrotron-based X-ray spectroscopy has proven to be valuable to study spin and orbital moments of transition metals and lanthanides in technologically relevant thin-film and monolayer samples. The access to coherent spin superposition states relevant for emergent quantum technologies remains, however, elusive with ordinary X-ray spectroscopy. Here, we approach the study of such quantum-coherent states via the X-ray detection of microwave-driven electron paramagnetic resonance, which involves much smaller signal levels than X-ray detected ferromagnetic resonance on classical magnets. We demonstrate the feasibility of this approach with thin films of phthalocyanine-based metal complexes containing copper or vanadium centers. We also identify X-ray specific phenomena that we relate to charge trapping of secondary electrons resulting from the decay of the X-ray excited core-hole state. Our findings pave the way toward the element-specific X-ray detection of coherent superposition states in monolayers of atomic and molecular spins on virtually arbitrary surfaces.