We determine the set of polynomials $f(x)\in k[x]$, where $k$ is a finite field, such that the local system on $\mathbb G_m^2$ which parametrizes the family of exponential sums $(s,t)\mapsto\sum_{x\in k}ψ(sf(x)+tx)$ has finite monodromy, in two cases: when $f(x)=x^d+λx^e$ is a binomial and when $f(x)=(x-α)^d(x-β)^e$ is of Belyi type.