Let k ≥ 2 and s be positive integers. Let θ ∈ (0, 1) be a real number. In this paper, we establish that if s > k(k + 1) and θ > 0.55, then every sufficiently large natural number n, subject to certain congruence conditions, can be written as n=p1k+⋯+psk, , where pi (1 ≤ i ≤ s) are primes in the interval ((ns)1k−nθk,(ns)1k+nθk] . The second result of this paper is to show that if s>k(k+1)2 and θ > 0.55, then almost all integers n, subject to certain congruence conditions, have the above representation.