If the voltage difference came from the circuit and not the solution, that would make sense. Think about how a galvanic cell's anode gets a negative charge in the first place. It's from the anions themselves and because of reactivity and solubility, not the charge.
If you consider an electrolytic cell, where the circuit produces its own sufficient voltage, everything is flipped. Basically instead of connecting the two electrodes with just a wire, you connect them with a wire and a battery or some voltage source. The orientation of the voltage source has to be such that it forces current in the direction opposite the normal galvanic cell. Ion migration switches, and you are recharging the chemical solution. Then negative ions will go the cathode.