Work function is a constant and has absolutely nothing to do with external field. It's the energy required to liberate. The only thing 0 sows that it was provided the exact work function on from light. I think it has something to do with. Gradient.
Yea you have to overcome work function to liberate electrons. It's the voltage that eliminated it (by giving the electrons more energy, if that makes sense). Hence you have to equate work function somehow with voltage. W=qV works perfectly. (they are also both in joules)
The gradient is planck's constant I'm not sure if you can find it that way.
Unless someone else found a better method, I'm sure this is the right one. I'm pretty sure that with a high voltage you can liberate elections without photoelectric effect. Again, shows that voltage does have something to do with electrons overcoming work function.
The circuit is also connected in such a way that the voltage gives the electrons a boost in energy. 4.1 volts is the amount of volts that photoelectric effect occurs at 0Hz. Shows that work function has just been overcome at this value.
I'm rambling but does that makes sense? :s