Embodiments of a method of removing carbonaceous deposits in a liquid-hydrocarbon fueled solid oxide fuel cell and related system are provided. The method includes providing a solid oxide fuel cell system having an anode, a cathode, a solid oxide electrolyte oriented between the anode and cathode, an amplifier cathode disposed proximate the solid oxide electrolyte and the cathode, a fuel cell electric circuit electrically connecting the anode and the cathode, and an amplifier electric circuit electrically connecting the anode and the amplifier cathode. Further, operating the amplifier electric circuit in an electrolytic mode to electrically power the amplifier cathode, wherein the amplifier cathode generates and supplies O2− or CO32− to the anode. The method further includes removing the carbonaceous deposits on the anode by converting the carbonaceous deposits to carbon dioxide gas via reaction with the O2− or CO32− and expelling the carbon dioxide gas.