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

Let k≥2 be a fixed natural number. We establish the existence of infinitely many pairs of consecutive primes pn, pn+1 satisfyingpn+1−pn≥clog⁡pnlog2⁡pnlog4⁡pnlog3⁡pn, with c being a fixed positive constant, for which the interval (pn,pn+1) contains the k-th power of a prime number.