  In this paper, we are concerned with the numerical solution for the
two-dimensional time fractional Fokker-Planck equation with tempered fractional
derivative of order $\alpha$. Although some of its variants are considered in
many recent numerical analysis papers, there are still some significant
differences. Here we first provide the regularity estimates of the solution.
And then a modified $L$1 scheme inspired by the middle rectangle quadrature
formula on graded meshes is employed to compensate for the singularity of the
solution at $t\rightarrow 0^{+}$, while the five-point difference scheme is
used in space. Stability and convergence are proved in the sence of
$L^{\infty}$ norm, then a sharp error estimate
$\mathscr{O}(\tau^{\min\{2-\alpha, r\alpha\}})$ is derived on graded meshes.
Furthermore, unlike the bounds proved in the previous works, the constant
multipliers in our analysis do not blow up as the Caputo fractional derivative
$\alpha$ approaches the classical value of 1. Finally, we perform the numerical
experiments to verify the effectiveness and convergence order of the presented
algorithms.
