  Suppose $X=\{X_t, t\ge 0\}$ is a supercritical superprocess. Let $\phi$ be
the non-negative eigenfunction of the mean semigroup of $X$ corresponding to
the 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 $M_\infty(\phi)$. In this paper we study 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 and necessary conditions for the rate in the almost sure
sense. Some results on the convergence rate in $L^p$ with $p\in(1, 2)$ are also
obtained.
