Is there a Context-free grammar for this language?
Here is a PDA for this language. While reading $x$, the whole of $x$ is pushed onto the stack. After encountering $\#$, the PDA pops some non-deterministic amount of symbols, then verifies that the top symbol is $1$, and pops $m$ symbols. It acceptes if the stack is empty.

Variant: instead of popping a part of $x$, while reading $x$ the PDA non-deterministically stops pushing symbols into the stack; it only does so after reading a $1$.

As for a CFG for this language, here is the idea:
$$
\begin{align*}
&amp;S \to \Sigma^{m-1} 1 T 1^m \\
&amp;T \to \Sigma^* \#
\end{align*}
$$

I'll let you construct an actual grammar based on this idea.