= Solution
Write $h_{ab}=\delta g_{ab}$ and $C^a{}_{bc}=\delta\Gamma^a{}_{bc}$. The difference between two <affine connections> is a tensor, so its infinitesimal change $C$ is a type $(1,2)$ <tensor field>. Both <affine connections> are <torsion-free>, giving $C^a{}_{bc}=C^a{}_{cb}$. Varying <metric compatibility> gives
$$
\nabla_c h_{ab}=g_{db}C^d{}_{ca}+g_{ad}C^d{}_{cb}.
$$
Lower the first index of $C$. Add the versions with derivatives $b,c$ and subtract the one with derivative $d$; the lower-slot symmetry cancels the unwanted terms, leaving
$$
\nabla_bh_{dc}+\nabla_ch_{db}-\nabla_dh_{bc}=2g_{da}C^a{}_{bc}.
$$
Therefore
$$
\boxed{\delta\Gamma^a{}_{bc}=\frac12g^{ad}
(\nabla_ch_{db}+\nabla_bh_{dc}-\nabla_dh_{bc}).}
$$
All <covariant derivatives> use the original <Levi-Civita connection>. The PDF has $\nabla_bh_{dc}$ as its second term; the converted TeX's $\nabla_dh_{dc}$ is a transcription error.
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