Square-root cancellation for sums of coefficients of $\mathrm{GL}(m)$-automorphic forms over values of random polynomials

For arbitrary $m\ge 2$ and $d\ge 1$, we establish square-root cancellation for sums of Fourier coefficients of $\mathrm{G}\mathrm{L}(m)$-automorphic forms averaged over values of random degree-$d$ polynomials.