Short intervals asymptotic formulae for binary problems with prime powers

Abstract: 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.