How to solve Poisson's equation with incomplete knowledge of the right-hand side?

Question

Consider the following Poission equation, $$\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\right)\varphi(x,y,z) = f(x,y,z).$$ I was wondering if we can calculate a plane of values for $\varphi(x,y,z)$ if we only know $f(x,y,z)$ at a plane of values? In other words, can we calculate $\varphi(x,y,z_0)$ if we know that $$f(x,y,z_0) = f_0(x,y)$$ but do not know the values of $f(x,y,z)$ $\forall z\ne z_0$?