Very few experiments probe quantum heat transport by photons, and all are made difficult by the forbidding use of a magnetic field to control heat flows. Here we present a magnetic field-free circuit where charge quantization in a superconducting island enables thorough electric field control. We thus tune the thermal conductance, close to its quantum limit, of a single photonic channel between two mesoscopic reservoirs. We observe heat flow oscillations originating from the competition between Cooper-pair tunnelling and Coulomb repulsion in the island, well captured by a simple model. Our results highlight the consequences of charge-phase conjugation on heat transport, with promising applications in thermal management of quantum devices and design of microbolometers. In addition, preliminary results indicate that the use of ultrasmall Josephson junctions coupled to high-impedance thermal baths leads to non-trivial thermal conduction mechanisms due to the bath back-acting on the junctions, providing an example of quantum thermal transport in strongly coupled systems.