In 2000, Baker, Harman and Pintz proved that there is always a prime in the interval $(n-n^{0.525}, n)$. There are also conditional results implying smaller intervals. Nevertheless, I could not find any information about prime power gaps. So, what I'm asking is:

What is the asymptotically largest function $f(n)$ s.t. there is always a prime power in the interval $(f(n), n)$?

For example, Bertrand’s postulate is almost trivial in this case, since there is always a power of $2$ in the interval $(n, 2n]$. On the other hand, the distribution of prime powers with exponent $e>1$ is much smaller than the distribution of primes, so adding them might not change the answer.