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

Let $k\ge 2$ be a fixed natural number. We establish the existence of infinitely many pairs of consecutive primes $p_n$, $p_{n+1}$ satisfying $$p_{n+1}-p_n\ge c\frac{\log p_n{\log }_2{p}_n{\log }_4{p}_n}{{\log }_3{p}_n},$$ with $c$ being a fixed positive constant, for which the interval $(p_n, p_{n+1})$containsthe$k$-th power of a prime number.