Sum of the triple divisor function of mixed powers

Let $d_3(n)$ denote the 3-th divisor function. In this paper, we study the asymptotic formula of the sum

$$\sum _{\begin{array}{c}1⩽n_1,n_2⩽X^{\frac{1}{2}}\\ 1⩽n_3⩽X^{\frac{1}{k}}\end{array}}{d}_3(n_1^2+n_2^2+n_3^k)$$

with $n_1,n_2,n_3\in {\mathbb{Z}}^{+}$ and $k⩾3$ be an integer. Previously only the case of $k=2$ is studied.