For fixed $k$, the existence of a prime power $p^k$ between $n$ and $2n$ is (asymptotically) equivalent to the existence of a prime $p$ between $m$ and $\sqrt[k]{2} m$ where $m = \sqrt[k]{n}$. Bachraoui gives an elementary proof here that there exists a prime between $m$ and $\frac{3}{2} m$ for sufficiently large $m$.

I remember reading on MO that it is known that similar elementary proofs exist for showing the existence of primes between $m$ and $(1 + \epsilon) m$ for any $\epsilon > 0$ and, furthermore, that the existence of these proofs itself depends on the PNT. However, I can't track down where I read this.