# Does photoelectric work function change with voltage?

Consider a typical photoelectric effect investigation setup like:

Light of a minimum frequency (threshold frequency) must be shone on the cathode in order to eject electrons. The minimum frequency corresponds to a certain energy of each photon by $$E=hf$$ and this energy represents the work function $$\phi$$ of the metal (minimum amount of energy needed to eject an electron). This produces a graph like this

However, what if you turned up the voltage in the experiment (such that the plate the electrons are flying towards - the anode - is positive)? Wouldn't this decrease the work function of the metal because the electric field is pulling on the electrons and thus less energy will be needed to push them out of the metal? Could someone tell me if my reasoning is correct or if there is no impact?