Grammar of words with exactly $k$ prefixes in another grammar

Question

Given a context-free grammar $G$, how can one systematically construct a grammar $G_k$ such that

$$ L(G_k) = \{w \in \Sigma^* : |\text{Pref}(w) \cap L(G)| = k\} $$

where $\text{Pref}(w)$ is the set of $|w|+1$ prefixes of $w$? Assume $G$ is unambiguous, if needed.