  We prove results about the asymptotic formulae in short intervals for the
average number of representations of integers of the forms
$n=p_{1}^{\ell_1}+p_{2}^{\ell_2}$, with $\ell_1, \ell_2\in\{2,3\}$,
$\ell_1+\ell_2\le 5$ are fixed integers, and $n=p^{\ell_1} + m^{\ell_2}$, with
$\ell_1=2$ and $2\le \ell_2\le 11$ or $\ell_1=3$ and $ \ell_2=2$ are fixed
integers,
  $p,p_1,p_2$ are prime numbers and $m$ is an integer.
