Pick elements that don't exhaust any set
This is the hitting set problem in disguise.
Let $U=\cup_{A_i \in \mathcal{C}} A_i$ be the universe and $\overline{X} = U \setminus X$.  Then your problem can be equivalently stated as follows: given $\mathcal{C}$ and $k$, find $\overline{X}$ that has $|U|-k$ elements, and such that $A_i \cap \overline{X} \ne \emptyset$ for each $A_i \in \mathcal{C}$.  That is exactly the hitting set problem.