Finite monodromy of some families of exponential sums

Given a prime $p$ and an integer $d>1$, we give a numerical criterion to decide whether the $\mathrm{\ell }$-adic sheaf associated to the one-parameter exponential sums $t\mapsto {\sum }_x\psi (x^d+tx)$ over ${\mathbb{F}}_p$ has finite monodromy or not, and work out some explicit cases where this is computable.