  In this paper, we study existence and multiplicity of solutions for the
following Kirchhoff-Choquard type equation involving the fractional
$p$-Laplacian on the Heisenberg group: \begin{equation*} \begin{array}{lll}
M(\|u\|_{\mu}^{p})(\mu(-\Delta)^{s}_{p}u+V(\xi)|u|^{p-2}u)=
f(\xi,u)+\int_{\mathbb{H}^N}\frac{|u(\eta)|^{Q_{\lambda}^{\ast}}}{|\eta^{-1}\xi|^{\lambda}}d\eta|u|^{Q_{\lambda}^{\ast}-2}u
&\mbox{in}\ \mathbb{H}^N, \\ \end{array} \end{equation*} where
$(-\Delta)^{s}_{p}$ is the fractional $p$-Laplacian on the Heisenberg group
$\mathbb{H}^N$, $M$ is the Kirchhoff function, $V(\xi)$ is the potential
function, $0<s<1$, $1<p<\frac{N}{s}$, $\mu>0$, $f(\xi,u)$ is the nonlinear
function, $0<\lambda<Q$, $Q=2N+2$, and
$Q_{\lambda}^{\ast}=\frac{2Q-\lambda}{Q-2}$ is the Sobolev critical exponent.
Using the Krasnoselskii genus theorem, the existence of infinitely many
solutions is obtained if $\mu$ is sufficiently large. In addition, using the
fractional version of the concentrated compactness principle, we prove that
problem has $m$ pairs of solutions if $\mu$ is sufficiently small. As far as we
know, the results of our study are new even in the Euclidean case.
