Certain floor function sums over powers

We study the sums $$S_f(x)=\sum _{n\le x}f\left(\lfloor \frac{x}{n}\rfloor \right)$$ when $f$ is supported on $r$th powers with $r\ge 2$. This restriction allows us to give nontrivial estimates for one of the error terms in the asymptotic expansion of $S_f(x)$. We also state several conjectures related to our results.