  In this article, we investigate the Kirchhoff-Schr\"{o}dinger-Poisson type
systems on the Heisenberg group of the following form: \begin{equation*}
\left\{ \begin{array}{lll} {-(a+b\int_{\Omega}|\nabla_{H}
u|^{p}d\xi)\Delta_{H,p}u-\mu\phi |u|^{p-2}u}=\lambda
|u|^{q-2}u+|u|^{Q^{\ast}-2}u &\mbox{in}\ \Omega, \\ -\Delta_{H}\phi=|u|^{p}
&\mbox{in}\ \Omega, \\ u=\phi=0 &\mbox{on}\ \partial\Omega, \end{array} \right.
\end{equation*} where $a,b$ are positive real numbers, $\Omega\subset
\mathbb{H}^N$ is a bounded region with smooth boundary, $1<p<Q$, $Q = 2N + 2$
is the homogeneous dimension of the Heisenberg group $\mathbb{H}^N$,
$Q^{\ast}=\frac{pQ}{Q-p}$, $q\in(2p, Q^{\ast})$, and
$\Delta_{H,p}u=\mbox{div}(|\nabla_{H} u|^{p-2}\nabla_{H} u)$ is the
$p$-horizontal Laplacian. Under some appropriate conditions for the parameters
$\mu$ and $\lambda$, we establish existence and multiplicity results for the
system above. To some extent, we generalize the results of An and Liu (Israel
J. Math., 2020) and Liu et al. (Adv. Nonlinear Anal., 2022).
