  We prove a recent conjecture due to Cluckers and Veys on exponential sums
modulo $p^m$ for $m \geq 2$ in the special case where the phase polynomial $f$
is sufficiently non-degenerate with respect to its Newton polyhedron at the
origin. Our main auxiliary result is an improved bound on certain related
exponential sums over finite fields. This bound can also be used to settle a
conjecture of Denef and Hoornaert on the candidate-leading Taylor coefficient
of Igusa's local zeta function associated to a non-degenerate polynomial, at
its largest non-trivial real candidate pole.
