In the present work, we are concerned with the existence of weak solutions for the Kirchhoff type double-phase problem
\begin{align}
\tag{$P_{\lambda}$}
\begin{split}
-M\left[\varphi_{\mathcal{H}}(\lvert \nabla u\rvert)\right] \mathcal{L}_{p(x),q(x)}^{\mu(x)}(u)+V(x)( \lvert u\rvert^{p(x)-2}u+\mu(x)\lvert u\rvert^{q(x)-2}u)\\=\lambda w(x) \lvert u\rvert^{s(x)-2}u -h(x)\lvert u\rvert^{r(x)-2}u,
\end{split}
\end{align}
where
\begin{align*}
\varphi_{\mathcal{H}}(u)=&\int_{\mathbb{R}^N}\left(\frac{|u|^{p(x)}}{p(x)}+\mu(x)\frac{| u|^{q(x)}}{q(x)}\right)\,dx,\\
\mathcal{L}_{p(x),q(x)}^{\mu(x)}(u)=&\text{div}(\lvert \nabla u\rvert^{p(x)-2} \nabla u+\mu(x)\lvert \nabla u\rvert^{q(x)-2} \nabla u),
\end{align*}
$ 0 \leq \mu(\cdot)\in L^1(\mathbb{R}^N)$, $\lambda$ is a real parameter and $V$ is an unbounded potential such that $V(x)\geq V_0>0$. The main distinctive feature of this problem lies in the second nonlinearity on the right-hand side, which can be in the supercritical. Additionally, we encounter the Kirchhoff term, which might become zero at the origin. By imposing certain assumptions, we establish the existence and multiplicity results.