Let $R(N)$ be the number of representations of $N$ as a sum of a prime and a square-full number weighted with logarithmic function. In $2024$, the author and Y. Suzuki obtained an asymptotic formula for the sum of $R(N)$ over positive integers $N$ in a short interval ($X$, $X+H$] for $X^{ \frac{ 1}{ 2}+\varepsilon} \le H < X^{ 1-\varepsilon}$. In this article, we improve the range of $H$, that is, we prove the same asymptotic formula for $X^{ \frac{ 32-4\sqrt{ 15}}{ 49}+\varepsilon} \le H \le X^{ 1- \varepsilon}$.
Categories: math.NT