Generalized mean value property for the Poisson equation
Solution 1:
Note that in your wish equality would imply that if $u$ is constant on a sphere, it is also constant on the whole ball. This is certainly not true.
I think what you're looking for is the Feynman-Kac formula, which says $$ u(x)=\frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)}u \;dS+\int_{B(x,r)}f\;d\mu, $$ where the measure $\mu$ can be expressed by a Brownian motion $W$ and its exit time $\tau$ from the ball of radius $r$: $$ \int_{B(x,r)}f\;d\mu=\mathbb{E}\int_0^{\tau}f(x+W_t)\;dt. $$