Transition discs provide an important tool to probe various mechanisms that might influence the evolution of protoplanetary discs and therefore the formation of planetary systems. One of these mechanisms is photoevaporation due to energetic radiation from the central star, which can in principal explain the occurrence of discs with inner cavities. Current models, however, fail to reproduce a subset of the observed transition discs, namely objects with large measured cavities and vigorous accretion. For these objects the presence of (multiple) giant planets is often invoked to explain the observations. In our work, we explore the possibility of X-ray photoevaporation operating in discs with different gas-phase depletion of carbon and show that the influence of photoevaporation can be extended in such low-metallicity discs. As carbon is one of the main contributors to the X-ray opacity, its depletion leads to larger penetration depths of X-rays in the disc and results in higher gas temperatures and stronger photoevaporative winds. We present radiation-hydrodynamical models of discs irradiated by internal X-ray+EUV radiation assuming carbon gas-phase depletions by factors of 3,10 and 100 and derive realistic mass-loss rates and profiles. Our analysis yields robust temperature prescriptions as well as photoevaporative mass-loss rates and profiles which may be able to explain a larger fraction of the observed diversity of transition discs.