= Solution
By translation and rotation invariance, use a radial <Green function> $G(R)$, where $R=|\mathbf r-\mathbf r'|$ is the <Euclidean distance>. Away from the source, the radial <modified Helmholtz equation> is
$$
G''+\frac2R G'-k^2G=0.
$$
Writing $u=RG$ reduces this to $u''-\kappa^2u=0$, where $\kappa=|k|$. For $\kappa>0$, decay eliminates $e^{\kappa R}/R$, leaving $G=Ae^{-\kappa R}/R$. For $\kappa=0$, the two radial branches are a constant and $1/R$; decay again leaves $A/R$.
To fix the singular normalization, integrate the defining equation over a ball of radius $\varepsilon$ about $\mathbf r'$. The <divergence theorem> gives
$$
4\pi\varepsilon^2G'(\varepsilon)-k^2\int_{R<\varepsilon}G\,d^3\mathbf r=1.
$$
The volume term is $O(\varepsilon^2)$, whereas
$$
4\pi\varepsilon^2G'(\varepsilon)=-4\pi A e^{-\kappa\varepsilon}(1+\kappa\varepsilon)\longrightarrow-4\pi A.
$$
Consequently
$$
\boxed{A=-\frac1{4\pi},\qquad G(\mathbf r,\mathbf r')=-\frac{e^{-|k||\mathbf r-\mathbf r'|}}{4\pi|\mathbf r-\mathbf r'|}.}
$$
\b[Source qualification.] The printed $e^{-kR}$ is the decaying branch only when $k\geq0$, although the source declares $k$ real. For negative $k$ it grows, so it cannot be the requested decaying <Green function>. The absolute value above covers every real $k$ and reduces to the printed form for nonnegative $k$.
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