$p$-Hitting-Set is W[2]-complete under fpt-reductions [FG06, Theorem 7.14]. It is in W[2] ($WH_E1_HittingSetInW2$), and it is W[2]-hard in two steps.

1. Every $\Pi_2$ weighted definability problem reduces to weighted monotone CNF satisfiability [FG06, Theorem 7.1]. Let $\varphi = \forall \bar x\, \exists \bar y\, \psi(X)$ and let $(\mathcal A, k)$ be an instance. As in $WH_D07_DefinabilityToWSat$, the universe is restricted to a set $U$ of polynomially many elements. A witness is an increasing list $t_0 < \dots < t_{k-1}$ of codes of $s$-tuples over $U$. The propositional variables describe, for each of $(k+1)^D$ blocks, the values of $D$ positions of that list, $D$ depending on $\psi$ only. Monotone clauses force exactly one value per block and the consistency of any two blocks, so that the true variables describe a single list; and for every assignment to $\bar x$, one clause collects the block values under which some assignment to $\bar y$ makes $\psi$ true. The weight is the number of blocks. This monotone formula replaces the propositional normalization of [FG06, Lemma 7.5].
2. Weighted monotone CNF satisfiability reduces to Hitting Set [FG06, Theorem 7.14]: the hyperedges are the clauses, read as sets of variables; an assignment of weight $k$ satisfies the formula exactly when its true variables meet every clause.