It is well known that for a stochastic differential equation driven by Lévy noise, the temporal Hölder continuity in $L^p$ sense of the exact solution does not exceed $1/p$. This leads to that the $L^p$-strong convergence order of a numerical scheme will vanish as $p$ increases to infinity if the temporal Hölder continuity of the solution process is directly used. A natural question arises: can one obtain the $L^p$-strong convergence order that does not depend on $p$? In this paper, we provide a positive answer for fully discrete schemes of the stochastic partial differential equation (SPDE) driven by Lévy noise. Two cases are considered: the first is the linear multiplicative Poisson noise with $ν(χ)<\infty$ and the second is the additive Poisson noise with $ν(χ)\leq\infty$, where $ν$ is the Lévy measure and $χ$ is the mark set. For the first case, we present a strategy by employing the jump-adapted time discretization, while for the second case, we introduce the approach based on the recently obtained Lê's quantitative John--Nirenberg inequality. We show that proposed schemes converge in $L^p$ sense with orders almost $1/2$ in both space and time for all $p\ge2$, which contributes novel results in the numerical analysis of the SPDE driven by Lévy noise.