Certain floor function sums over powers

We study the sums $$S_f(x) = \sum_{n\leq x} f\left(\left\lfloor\frac{x}{n}\right\rfloor\right)$$ when $f$ is supported on $r$th powers with $r\geq 2$. 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.