After reading several posts and sources online, I think I finally understand the saturation current: at the saturation current, all of the electrons being emitted due to the incident light are being collected by the anode.
However what I do not understand is this: whatever current, frequency, and intensity of incident light I am shining on the cathode, there will be electrons which are absorbing some photons but are too deep in the metal to come out. From what I understand the calculation of current for some constant anode potential takes into account this already, that each electron that contributes to the current could be bring released after absorbing several photons.
However, I was also wondering about increasing the anode voltage at this point. Should it not be the case that some of these extra 'stuck' electrons will be dislodged, and furthermore could I not increase the voltage sufficiently to overcome the work function so that the electrons from the cathode jump to the anode even without incident light?
If this is the case I do not see why a saturation current exists. I understand it in terms of 'collecting all of the electrons released by the incident light', but I do not understand why increasing the voltage does not itself dislodge more electrons, thereby increasing the current.