Let $$\mathcal {H}=(V,\mathcal {E})$$ H = ( V , E ) be a hypergraph with set of vertices $$V, n:=|V|$$ V , n : = | V | and set of (hyper-)edges $$\mathcal {E}, m:=|\mathcal {E}|$$ E , m : = | E | . Let $$l$$ l be the maximum size of an edge, $$\varDelta $$ Δ be the maximum vertex degree and $$D$$ D be the maximum edge degree. The $$k$$ k -partial vertex cover problem in hypergraphs is the problem of finding a minimum cardinality subset of vertices in which at least $$k$$ k hyperedges are incident. For the case of $$k=m$$ k = m and constant $$l$$ l it known that an approximation ratio better than $$l$$ l cannot be achieved in polynomial time under the unique games conjecture (UGC) (Khot and Ragev J Comput Syst Sci, 74(3):335–349, 2008), but an $$l$$ l -approximation ratio can be proved for arbitrary $$k$$ k (Gandhi et al. J Algorithms, 53(1):55–84, 2004). The open problem in this context has been to give an $$\alpha l$$ α l -ratio approximation with $$\alpha < 1$$ α < 1 , as small as possible, for interesting classes of hypergraphs. In this paper we present a randomized polynomial-time approximation algorithm which not only achieves this goal, but whose analysis exhibits approximation phenomena for hypergraphs with $$l\ge 3$$ l ≥ 3 not visible in graphs: if $$\varDelta $$ Δ and $$l$$ l are constant, and $$2\le l\le 4\varDelta $$ 2 ≤ l ≤ 4 Δ , we prove for $$l$$ l -uniform hypergraphs a ratio of $$l\left( 1-\frac{l-1}{4\varDelta }\right) $$ l 1 - l - 1 4 Δ , which tends to the optimal ratio 1 as $$l\ge 3$$ l ≥ 3 tends to $$4\varDelta $$ 4 Δ . For the larger class of hypergraphs where $$l, l \ge 3$$ l , l ≥ 3 , is not constant, but $$D$$ D is a constant, we show a ratio of $$l(1-1/6D)$$ l ( 1 - 1 / 6 D ) . Finally for hypergraphs with non-constant $$l$$ l , but constant $$\varDelta $$ Δ , we get a ratio of $$ l (1- \frac{2 - \sqrt{3}}{6\varDelta })$$ l ( 1 - 2 - 3 6 Δ ) for $$k\ge m/4$$ k ≥ m / 4 , leaving open the problem of finding such an approximation for $$k < m/4$$ k < m / 4 .