  The discovery that a gate electrode suppresses the supercurrent in purely
metallic systems is missing a complete physical understanding of the mechanisms
at play. We here study the origin of this reduction in a Superconductor-Normal
metal-Superconductor Josephson junction by performing, on the same device, a
detailed investigation of the gate-dependent switching probability together
with the local tunnelling spectroscopy of the normal metal. We demonstrate that
high energy electrons leaking from the gate trigger the reduction of the
critical current which is accompanied by an important broadening of the
switching histograms. The switching rates are well described by an activation
formula including an additional term accounting for the injection of rare high
energy electrons from the gate. The rate of electrons obtained from the fit
remarkably coincides with the independently measured leakage current.
Concomitantly, a negligible elevation of the local temperature is found by
tunnelling spectroscopy which excludes overheating scenarios.
