Short intervals asymptotic formulae for binary problems with prime powers

We prove results about the asymptotic formulae in short intervals for the average number of representations of integers of the forms $n=p_1^{{\mathrm{\ell }}_1}+p_2^{{\mathrm{\ell }}_2}$, with ${\mathrm{\ell }}_1,{\mathrm{\ell }}_2\in \{2,3\}$, ${\mathrm{\ell }}_1+{\mathrm{\ell }}_2\le 5$ are fixed integers, and $n=p^{{\mathrm{\ell }}_1}+m^{{\mathrm{\ell }}_2}$, with ${\mathrm{\ell }}_1=2$ and $2\le {\mathrm{\ell }}_2\le 11$ or ${\mathrm{\ell }}_1=3$ and ${\mathrm{\ell }}_2=2$ are fixed integers, $p,p_1,p_2$ are prime numbers and $m$ is an integer.