This $\partial_i$ is used by Hamilton to denote the Levi-Civita connection with respect to the metric $g$, or with respect to $\Gamma_{i}{}^{k}_{j}$
Notice that there is a typo in the original paper in the fifth line on p.52 where it is said that "$\partial_i$ is covariant differentiation with respect to $F_{i}{}^{k}_{j}$".
Let me change the notation to make the things clearer. I use $\nabla^{(g)}$ and $\nabla^{(h)}$ for the Levi-Civita connections of the metrics $g$ and $h$ respectively.
In other words, I use $\nabla^{(g)}_i$ instead of Hamilton's $\partial_i$ to avoid a possible confusion with the partial derivatives.
The difference of the connections $F_{i}{}^{k}_{j}$ in then expressed by the identity:
$$
\nabla^{(g)}_i \omega_j = \nabla^{(h)}_i \omega_j + F_{i}{}^{k}_{j} \omega_k
$$
where $\omega_i$ is an arbitrary $1$-form (covector).
Now we can calculate the Laplacian w.r.t. $g$ of the function $E = g^{i j} h_{i j}$.
$$
\begin{align}
\Delta^{(g)} E & = g^{i j} \nabla^{(g)}_i \nabla^{(g)}_j (g^{k l} h_{k l}) \\
& = g^{i j} g^{k l} \nabla^{(g)}_i \nabla^{(g)}_j h_{k l} \\
& = g^{i j} g^{k l} \nabla^{(g)}_i \Big( \nabla^{(h)}_j h_{k l} + F_{j}{}^{p}_{k} h_{p l} + F_{j}{}^{p}_{l} h_{k p} \Big) \
\end{align}
$$
Now we observe that $\nabla^{(h)}_j h_{k l} = 0$ and recover the identity
$$
\nabla^{(g)}_j h_{k l} = F_{j}{}^{p}_{k} h_{p l} + F_{j}{}^{p}_{l} h_{k p}
$$
Continuing this process we obtain
$$
\begin{align}
\Delta^{(g)} E & = g^{i j} g^{k l} \nabla^{(g)}_i \Big( F_{j}{}^{p}_{k} h_{p l} + F_{j}{}^{p}_{l} h_{k p} \Big) \\
& = g^{i j} g^{k l} \Big( h_{p l} \nabla^{(g)}_i F_{j}{}^{p}_{k} + F_{j}{}^{p}_{k} \nabla^{(g)}_i h_{p l} + h_{k p} \nabla^{(g)}_i F_{j}{}^{p}_{l} + F_{j}{}^{p}_{l} \nabla^{(g)}_i h_{k p} \Big) \\
& = 2 \, g^{i j} g^{k l} h_{p l} \nabla^{(g)}_i F_{j}{}^{p}_{k} + 2\, g^{i j} g^{k l} F_{j}{}^{p}_{k} \nabla^{(g)}_i h_{p l}
\end{align}
$$
which is equivalent to the equation in the question.
This also confirms the identity in @Avitus's answer:
$$
\begin{align}
\Delta^{(g)} E & = 2 \, g^{i j} g^{k l} \nabla^{(g)}_i \Big( h_{p l} F_{j}{}^{p}_{k} \Big)
\end{align}
$$
