Short intervals asymptotic formulae for binary problems with prime powers, II

Abstract: We improve some results about the asymptotic formulae in short intervals for the average number of representations of integers of the forms $n=p_{1}^{\ell_1}+p_{2}^{\ell_2}$ and $n=p^{\ell_1} + m^{\ell_2}$, where $\ell_1, \ell_2\ge 2$ are fixed integers, $p,p_1,p_2$ are prime numbers and $m$ is an integer.