Asymptotic formula for the sum of a prime and a square-full number in short intervals

Let $R_{m,\mathrm{s}\mathrm{q}-\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}(N)$ be a representation function for the sum of a prime and a square-full number. In this article, we prove an asymptotic formula for the sum of $R_{m,\mathrm{s}\mathrm{q}-\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}(N)$ over positive integers $N$ in a short interval ($X$, $X+H$] of length $H$ slightly bigger than $X^{\frac{1}{2}}$.