As shown in the figure, after adding a small perturbation term $h$ to the metric $g$, the expression of Christoffel symbols in an article is Eq. (10)
But I think the result should be in the following form. Is my idea correct?
References: https://doi.org/10.1103/PhysRev.108.1063 eq. (5).
You're technically correct here, but you also have to remember that $\bar{g}_{\alpha \beta;\nu} = 0$ by the definition of the covariant derivative. So all of your red terms vanish.
(I'm assuming here that the ${}_{;\nu}$ subscript notation refers to the covariant derivative associated with the background metric, which is how these calculations normally proceed. If it somehow refers to the covariant derivative associated with the perturbed metric, then $\bar{g}_{\alpha \beta;\nu}$ is not zero, but it is of $\mathcal{O}(h)$ and so it can be ignored at linear order in perturbation theory.)
I think your confusion lies in the distinction between "," and ";", i.e. partial and covariant derivative. First of all, by the product rule, we have that
$$\delta\Gamma^{\alpha}_{\mu\nu}=\delta\bigg(\frac{1}{2}g^{\alpha\delta}(\partial_{\mu}g_{\nu\delta}+\partial_{\nu}g_{\mu\delta}-\partial_{\delta}g_{\mu\nu})\bigg)=-\frac{1}{2}h^{\alpha\delta}(\partial_{\mu}\overline{g}_{\nu\delta}+\partial_{\nu}\overline{g}_{\mu\delta}-\partial_{\delta}\overline{g}_{\mu\nu})+\frac{1}{2}\overline{g}^{\alpha\delta}(\partial_{\mu}h_{\nu\delta}+\partial_{\nu}h_{\mu\delta}-\partial_{\delta}h_{\mu\nu})=-\frac{1}{2}h^{\alpha\delta}(\overline{g}_{\nu\delta,\mu}+\overline{g}_{\mu\delta,\nu}-\overline{g}_{\mu\nu,\delta})+\frac{1}{2}\overline{g}^{\alpha\delta}(h_{\nu\delta,\mu}+h_{\mu\delta,\nu}-h_{\mu\nu,\delta})$$
Note that the linearization of $g$ with upper indices is $g^{\mu\nu}=\overline{g}^{\mu\nu}-h^{\mu\nu}$. This is where the initial minus come from. Next, observe that
$$\delta\Gamma^{\alpha}_{\mu\nu}=-\frac{1}{2}h^{\alpha\delta}(\partial_{\mu}\overline{g}_{\nu\delta}+\partial_{\nu}\overline{g}_{\mu\delta}-\partial_{\delta}\overline{g}_{\mu\nu})+\frac{1}{2}\overline{g}^{\alpha\delta}(\partial_{\mu}h_{\nu\delta}+\partial_{\nu}h_{\mu\delta}-\partial_{\delta}h_{\mu\nu})=\frac{1}{2}\overline{g}^{\alpha\delta}(\nabla_{\mu}h_{\nu\delta}+\nabla_{\nu}h_{\mu\delta}-\nabla_{\delta}h_{\mu\nu})=\frac{1}{2}\overline{g}^{\alpha\delta}(h_{\nu\delta;\mu}+h_{\mu\delta;\nu}-h_{\mu\nu;\delta})$$
where the covariant derivatives are with respect to the background metric $g$. To see the last to last equality, just spell out the terms $\nabla_{\mu}h_{\nu\delta}$ etc. explicitely using Christoffel symbols.
