On the Waring-Goldbach problem on average

Let $s$, $\mathrm{\ell }$ be two integers such that $2\le s\le \mathrm{\ell }-1$, $\mathrm{\ell }\ge 3$. We prove that a suitable asymptotic formula for the average number of representations of integers $n={\sum }_{i=1}^s{p}_i^{\mathrm{\ell }}$, where $p_i$, $i=1,\dots ,s$, are prime numbers, holds in short intervals.