  In this article, we study the semi discrete and fully discrete formulations
for a Kirchhoff type quasilinear integro-differential equation involving
time-fractional derivative of order $\alpha \in (0,1) $. For the semi discrete
formulation of the equation under consideration, we discretize the space domain
using a conforming FEM and keep the time variable continuous. We modify the
standard Ritz-Volterra projection operator to carry out error analysis for the
semi discrete formulation of the considered equation. In general, solutions of
the time-fractional partial differential equations (PDEs) have a weak
singularity near time $t=0$. Taking this singularity into account, we develop a
new linearized fully discrete numerical scheme for the considered equation on a
graded mesh in time. We derive a priori bounds on the solution of this fully
discrete numerical scheme using a new weighted $H^{1}(\Omega)$ norm. We prove
that the developed numerical scheme has an accuracy rate of
$O(P^{-1}+N^{-(2-\alpha)})$ in $L^{\infty}(0,T;L^{2}(\Omega))$ as well as in
$L^{\infty}(0,T;H^{1}_{0}(\Omega))$, where $P$ and $N$ are degrees of freedom
in the space and time directions respectively. The robustness and efficiency of
the proposed numerical scheme are demonstrated by some numerical examples.
