  In this paper, we show that under a mild condition, a principal submodule of
the Bergman module on a bounded strongly pseudoconvex domain with smooth
boundary in $\mathbb{C}^n$ is $p$-essentially normal for all $p>n$. This
improves a previous result by the first author and K. Wang, in which it was
shown that any polynomial-generated principal submodule of the Bergman module
on the unit ball $\mathbb{B}_n$ is $p$-essentially normal for all $p>n$. As a
consequence, we show that the submodule of $L_a^2(\mathbb{B}_n)$ consisting of
functions vanishing on an analytic subset of pure codimension $1$ is
$p$-essentially normal for all $p>n$.
