  Let $(\Omega,\Sigma,\mu)$ be a complete probability space, $X$ a Banach space
and $1\leq p<\infty$. In this paper we discuss several aspects of $p$-Dunford
integrable functions $f:\Omega \to X$. Special attention is paid to the
compactness of the Dunford operator of $f$. We also study the $p$-Bochner
integrability of the composition $u\circ f:\Omega \to Y$, where $u$ is a
$p$-summing operator from $X$ to another Banach space $Y$. Finally, we also
provide some tests of $p$-Dunford integrability by using $w^*$-thick subsets of
$X^*$.
