Large gaps between consecutive prime numbers containing perfect $k$-th powers of prime numbers

Abstract: Let $k\geq 2$ be a fixed natural number. We establish the existence of infinitely many pairs of consecutive primes $p_n$, $p_{n+1}$ satisfying $$ p_{n+1}-p_n\geq c:\frac{\log p_n: \log_2 p_n: \log_4 p_n}{\log_3 p_n}:,$$ with $c$ being a fixed positive constant, for which the interval $(p_n, p_{n+1})$ contains the $k$-th power of a prime number.