This inequality (as it is formulated) is not true in the general case. Here is an example in two dimensions. Your notation $\overline{u}(r,w)$ is not proper because the average of $u(r,w)$ over the sphere does not depend on $w$. Let us define $$u(r,\phi):=\begin{cases} 1,\,\phi \ge 0, \phi \le \epsilon;\\ 0, \it{otherwise.}\\ \end{cases}$$ Then $\overline{u}(r)= \frac \epsilon {2\pi}.$ Therefore, the constant $C=\frac {2\pi} \epsilon,$ depending on $u$. Such inequality may be true under additional assumptions on $u$. For instance, it is true with the constant $C=1$ for subharmonic functions.

This inequality (as it is formulated) is not true in the general case. Here is an example in two dimensions. Your notation $\overline{u}(r,w)$ is not proper because the average of $u(r,w)$ over the sphere does not depend on $w$. Let us define $$u(r,\phi):=\begin{cases} 1,\mbox{ for }\phi \ge 0 \mbox{ and }\phi \le \epsilon,\\ 0, \mbox{ otherwise.}\\ \end{cases}$$ Then $\overline{u}(r)= \frac \epsilon {2\pi}.$ Therefore, the best possible constant $C$ is $\frac {2\pi} \epsilon$, and depends on $u$. Such inequality may be true under additional assumptions on $u$. For instance, it is true with the constant $C=1$ for subharmonic functions.