  Let $3\le d\le k$ and $\nu\ge 0$ be fixed and
$\mathcal{F}\subset\binom{[n]}{k}$. The matching number of $\mathcal{F}$,
denoted by $\nu(\mathcal{F})$, is the maximum number of pairwise disjoint sets
in $\mathcal{F}$, and $\mathcal{F}$ is $d$-cluster-free if it does not contain
$d$ sets with the union of size at most $2k$ and empty intersection. In this
paper, we give a lower bound and an upper bound for the maximum size of a
$d$-cluster-free family with a matching number at least $\nu+1$. In particular,
our result of the case $\nu=1$ settles a conjecture of Mammoliti and Britz. We
also introduce a Tur\'{a}n problem in hypergraphs that allows multiple edges,
which may be of independent interest.
