= Solution
With the positive logarithmic fundamental solution used here, a <Dirichlet Green function> for an interior pole $\mathbf r_0$ satisfies
$$
\nabla_{\mathbf r}^2G(\mathbf r,\mathbf r_0)=\delta(\mathbf r-\mathbf r_0),\qquad G=0\text{ on }\partial\mathcal D.
$$
Equivalently, $G-G_0$ is harmonic near the pole, $G$ is harmonic away from it, and it has the specified logarithmic singularity. Regularity up to the remaining boundary is understood; on an unbounded domain impose the appropriate condition at infinity as well.
Apply <Green second identity> with $f=\psi$ and $g=G$. Since $\psi$ is a <harmonic function>, the volume integral is $\int\psi\delta_{\mathbf r_0}\,dA=\psi(\mathbf r_0)$. The term $G\partial_n\psi$ vanishes on the boundary by the <Dirichlet boundary condition>. Therefore
$$
\boxed{\psi(\mathbf r_0)=\int_{\partial\mathcal D}\psi(\mathbf r)\,\partial_nG(\mathbf r,\mathbf r_0)\,dl.}
$$
One may equivalently excise a small disc around the pole and take its radius to zero. The sign is positive because this question uses $\nabla^2G=+\delta$, not the alternative negative-Laplacian convention.
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