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