  Let $R_{k,\ell}(N)$ be the representation function for the sum of the $k$-th
power of a prime and the $\ell$-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
$H=X^{\frac{1}{2}+\epsilon}$ 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
$(k,\ell)$. We also consider this general case, and as a corollary, we prove a
conditional result of Languasco and Zaccagnini (2018) for the case $\ell=2$
unconditionally up to some small factors.
