  Let $k \ge 2$ and $s$ be positive integers, and let $n$ be a large positive
integer subject to certain local conditions. We prove that if $s \ge k^2+k+1$
and $\theta > 31/40$, then $n$ can be expressed as a sum $p_1^k + \dots +
p_s^k$, where $p_1, \dots, p_s$ are primes with $|p_j - (n/s)^{1/k}| \le
n^{\theta/k}$. This improves on earlier work by Wei and Wooley and by Huang who
proved similar theorems when $\theta > 19/24$.
