Suppose $X=\{X_t, t\ge 0\}$ is a supercritical superprocess starting from a finite measure $\mu$. Let $\phi$ be the eigenfunction of the mean semigroup of $X$ corresponding to principal eigenvalue $\lambda>0$. Then $M_t(\phi)=e^{-\lambda t}\langle\phi, X_t\rangle, t\geq 0,$ is a non-negative martingale with almost sure limit $M_\infty(\phi)$. I will talk about the rate at which $M_t(\phi)-M_\infty(\phi)$ converges to $0$ as $t\to \infty$ when the process may not have finite variance. Under some conditions on the mean semigroup, we provide sufficient conditions and necessary conditions for the rate in almost sure sense. Some results on convergence rate in $L^p$ with $p\in (1, 2)$ are also obtained. The talk is based on a joint work with Rongli Liu and Renming Song.