When an un-gated thermionic cathode is operated in a radio-frequency gun, some fraction of the emitted electrons will return to the cathode due to the change in sign of the electric field in the gun. This back-bombardment current causes heating of the cathode, and this reduces the ability of the cathode heater to control the bunch charge. In this paper, we investigate the fundamental factors in single-frequency back-bombardment for a short-gap electron gun. Simulations revealed that the back-bombardment power depends strongly on the operating frequency and the bunch charge. Additionally, the use of a two-frequency electron gun to mitigate this effect was investigated which revealed that the effectiveness of this technique depends strongly upon single-frequency back-bombardment power but with the optimal reduction in back-bombardment power (62% of the baseline) occurring in the low-frequency limit.