  For "almost all" sufficiently large $N,$ satisfying necessary congruence
conditions and $k\geq 2$, we show that
  there is an {\bf asymptotic formula} for the number of solutions of the
equation \begin{align*} \begin{split} &N=p_{1}^{k}+p_{2}^{k}+\cdots+p_{s}^{k},
\\ &\left|p_{i}-( N/s)^{1/k}\right|\leq (N/s)^{\theta/k},\ (1\leq i\leq s)
\end{split} \end{align*} with \begin{align*} s\geq \frac{k(k+1)}{2}+1\ \
\textup{and}\ \ \theta\geq
  {\bf 2/3}+\varepsilon. \end{align*}
  This enlarges the effective range of $s$ for which can be obtained by the
method of M\"{a}tomaki and Xuancheng Shao \cite{MS}. [Discorrelation between
primes in short intervals and polynomial phase, Int. Math. Res. Not. IMRN 2021,
no. 16, 12330-12355.]
  The idea is to avoid using the exponential sums (1.2) and Vinogradov mean
value theorems in Lemma 2.4 simultaneously. And the main new ingredient is from
Kumchev and Liu \cite{KL} (see Lemma 2.2).
