On $$\mathrm GL_3$$ 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 $$\mathrm SL_3(\mathbb {Z})$$ and $$ \omega (x)$$ be a smooth compactly supported function. In this paper, we prove a nontrivial upper bound for the sum $$ \sum _{\begin{array}{c} n_1,\cdots ,n_\ell ,n_{\ell +1}\in \mathbb {Z}_+ \\ n=n_1^r+\cdots +n_{\ell }^r+n_{\ell +1}^s \end{array}} A_{\pi }(n,1)\omega \left( n/X\right) , $$ where $$r\ge 2$$ , $$s\ge 2$$ and $$\ell \ge 2^{r-1}$$ are integers.