How does an increase in battery voltage affect the maximum KE of an electron in a photoelectric cell?

In this case I will assume that $f$ and $\phi$ will remain constant since battery voltage is changing.

The two equations I have are

1) $KE_{max} = hf - \phi = h(f-f_0)$

and

2) $V_se = KE_{max}$

Initially my thought process was that $KE_{max}$ is determined from equation 1 assuming $f$ and $\phi$ remain constant. And because of that, any change in voltage would have no effect on the maximum kinetic energy.

However, another way I was thinking about it was that as the voltage increases the electrons will undergo more repulsion which lowers their kinetic energy.

Is one of these arguments more correct than the other? Or are they both factors in determining the effect on $KE_{max}$?