On certain kernel functions and shifted convolution sums of Hecke eigenvalues

Let $j\ge 2$ be a given integer. Let $f$ be a normalized primitive holomorphic cusp form of even integral weight for the full modular group $\mathrm{\Gamma }=SL(2,\mathbb{Z})$. Denote by ${\lambda }_{{\text{sym}}^{j}f}(n)$ the $n$th normalized coefficient of the Dirichlet expansion of the $j$th symmetric power $L$-function $L(s,{\text{sym}}^{j}f)$. In this paper, we are interested in the behavior of the shifted convolution sum involving ${\lambda }_{{\text{sym}}^{j}f}(n)$ with a weight function to be the $k$-full kernel function for any fixed integer $k\ge 2$.