Finite monodromy of some two-parameter families of exponential sums

We determine the set of polynomials f(x)∈ k[x], where k is a finite field, such that the local system on Gₘ² which parametrizes the family of exponential sums (s,t)↦∑x∈ kψ(sf(x)+tx) has finite monodromy, in two cases: when f(x)=xᵈ+λ xᵉ is a binomial and when f(x)=(x-α)ᵈ(x-β)ᵉ is of Belyi type.