PDA for a language where the second part is not the reverse of the first part
This answer is for the original version of the question, in which &quot;$k$&quot; was missing.
Your language contains all non-empty words. Indeed, if $w \neq \epsilon$ then you can write $w = nm$, where $m = \epsilon$ and $n = w \neq \epsilon = m^R$. In contrast, if $\epsilon = nm$ then $n=m=\epsilon$ and so necessarily $n=m^R$.