  We consider capacitated vertex cover with hard capacity constraints (VC-HC)
on hypergraphs. In this problem we are given a hypergraph $G=(V,E)$ with a
maximum edge size $f$. Each edge is associated with a demand and each vertex is
associated with a weight (cost), a capacity, and an available multiplicity. The
objective is to find a minimum-weight vertex multiset such that the demands of
the edges can be covered by the capacities of the vertices and the multiplicity
of each vertex does not exceed its available multiplicity.
  In this paper we present an $O(f)$ bi-approximation for VC-HC that gives a
trade-off on the number of augmented multiplicity and the cost of the resulting
cover. In particular, we show that, by augmenting the available multiplicity by
a factor of $k \ge 2$, a~cover with a cost ratio of
$\Big(1+\frac{1}{k-1}\Big)(f-1)$ to the optimal cover for the original instance
can be obtained. This improves over a previous result, which has a cost ratio
of $f^2$ via augmenting the available multiplicity by a factor of $f$.
