Is $\{ w \in L : wx \in L, \textrm{for some } x \neq \epsilon\}$ a CFG language if $L$ is CFG?
Consider the language
$$
L = \{ a^n b^m c^nd : n,m \geq 0 \} \cup \{ a^n b^m c^m d^2 : n,m \geq 0 \}.
$$
Then $\{ w \in L : wx \in L \text{ for some } x \neq \epsilon \} \cap a^* b^* c^* d = \{ a^n b^n c^n d : n \geq 0 \}$.

Here is a similar example, showing that if $L$ is context-free then $L \cap L \Sigma^+$ is not necessarily context-free:
$$
L = \{ a^n b^n : n \geq 0 \} \cup \{ a^n b^m c^m : n,m \geq 0 \}.
$$
Indeed, in this case $L \cap L\Sigma^+ = \{ a^n b^n c^n : n \geq 0 \}$.

The situation is likely different for deterministic context-free languages.