On $\mathrm{G}{\mathrm{L}}_3$ Fourier coefficients over values of mixed powers

Let $A_{\pi }(n,1)$ be the $(n,1)$-th Fourier coefficient of the Hecke-Maass cusp form $\pi$ for $\mathrm{S}{\mathrm{L}}_3(\mathbb{Z})$ and $\omega (x)$ be a smooth compactly supported function. In this paper, we prove a nontrivial upper bound for the sum $$\sum _{\genfrac{}{}{0px}{}{n_1,\cdots ,n_{\mathrm{\ell }},n_{\mathrm{\ell }+1}\in {\mathbb{Z}}_{+}}{n=n_1^r+\cdots +n_{\mathrm{\ell }}^r+n_{\mathrm{\ell }+1}^s}}{A}_{\pi }(n,1)\omega \left(n\mathrm{/}X\right),$$ where $r\ge 2$, $s\ge 2$ and $\mathrm{\ell }\ge 2^{r-1}$ are integers.