Big O notation and "input"
The asymptotic running time of your algorithm is $O(mn\log n)$, as you mention. If we don't know anything further about $m$ or $n$, there is no way to simplify this. If you know that $m = O(f(n))$, then the running time is also $O(f(n)n\log n)$. For example, if you know that $m = O(\sqrt{n})$, then the running time is also $O(n^{1.5}\log n)$. It will be $O(n\log n)$ only if $m = O(1)$, that is, $m$ is constant.