Inspired by a related question with respect to squarefree integers, I am asking how sharp one can obtain a result of the following type: let $k \geq 2$ be a fixed positive integer. For which functions $f : \mathbb{Z} \rightarrow \mathbb{R}$ can we conclude that intervals of the type $\left[n, n + \frac{f(n)\log n}{\log \log n}\right]$ contain no number that is $k$-free? That is, every number in the interval is divisible by $p^k$ for some prime $p$.
In this question (Consecutive non squarefree integers), the case $k=2$ is considered and that $f$ can be taken to be a constant.
Are there any non-trivial improvements (that is, not just an improvement on the constant $c$) for the case $k > 2$?