= Solution
Set $\kappa=|k|$. The differential operator in the <modified Helmholtz equation> depends only on $k^2=\kappa^2$. Let $G$ be its decaying <Green function>. Acting on the proposed integral with the operator, in the sense of <distributions>, gives
$$
\begin{aligned}
(\nabla_{\mathbf r}^2-k^2)\phi(\mathbf r)
&=\int_{\mathbb R^3}(\nabla_{\mathbf r}^2-k^2)G(\mathbf r,\mathbf r')\rho(\mathbf r')\,d^3\mathbf r'\\
&=\int_{\mathbb R^3}\delta^{(3)}(\mathbf r-\mathbf r')\rho(\mathbf r')\,d^3\mathbf r'\\
&=\rho(\mathbf r).
\end{aligned}
$$
The last step is the sifting property of the <Dirac delta>. Thus
$$
\boxed{\phi(\mathbf r)=\int_{\mathbb R^3}G(\mathbf r,\mathbf r')\rho(\mathbf r')\,d^3\mathbf r'.}
$$
For a regular rapidly decaying source, the $1/|\mathbf r-\mathbf r'|$ singularity of the kernel found in part (ii) is locally <integrable> in three dimensions, and its convolution with the source tends to zero at infinity. This verifies both the equation and the decay condition. The sign of the kernel below corresponds to $\nabla^2-k^2$; reversing the sign of the entire operator reverses the kernel.
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