On the sum of a prime power and a power in short intervals

Let $R_{k,\mathrm{\ell }}(N)$ be the representation function for the sum of the $k$-th power of a prime and the $\mathrm{\ell }$-th power of a positive integer. Languasco and Zaccagnini (2017) proved an asymptotic formula for the average of $R_{1,2}(N)$ over short intervals $(X,X+H]$ of the length $H$ slightly shorter than $X^{\frac{1}{2}}$, which is shorter than the length $H=X^{\frac{1}{2}+ϵ}$ in the exceptional set estimates of Mikawa (1993) and of Perelli and Pintz (1995). In this paper, we prove that the same asymptotic formula for $R_{1,2}(N)$ holds for $H$ of the size $X^{0.337}$. Recently, Languasco and Zaccagnini (2018) extended their result to more general $(k,\mathrm{\ell })$. We also consider this general case, and as a corollary, we prove a conditional result of Languasco and Zaccagnini (2018) for the case $\mathrm{\ell }=2$ unconditionally up to some small factors.