Distinction between hyper-Kloosterman sums and multiplicative functions

Let ${\text{kl}}_n(a,b;m)$ be the hyper-Kloosterman sum. Fix integers $n⩾2,a\mathrm{\ne }0$, $b\mathrm{\ne }0$ and $k⩾2$. For any $0\mathrm{\ne }\eta \in \mathbb{C}$ and multiplicative function $f:\mathbb{N}\to \mathbb{C}$, we prove that ${\text{kl}}_n(a,b;m)\mathrm{\ne }\eta f(m)$ holds for $100\mathrm{\%}$ square-free $k$-almost prime numbers $m$ and $100\mathrm{\%}$ square-free numbers $m$. Counterintuitively, if ${\text{kl}}_n(a,b;p)=\eta f(p)$ holds for all but finitely many primes $p$, we further show that \begin{align*} \ab|{m\leqslant X:\kl_n(a,b;m)=ηf(m), m $\text{ square-free }$k$\text{-almost prime}$}|= O(X^{1-\frac{1}{k+1}}). \end{align*} These results overturn the general belief that ${\text{kl}}_n(a,b;m)$ is nearly multiplicative in $m$, and that its distribution at almost prime moduli $m$ closely approximates that at primes. Moreover, we prove that these results also hold for general algebraic exponential sums satisfying some natural conditions.