Estimates are given for the exponential sum $\sum_{x=1}^p \exp(2\pi i f(x)/p)$, $p$ a prime and $f$ a nonzero integer polynomial, of interest in cases where the Weil bound is worse than trivial. The results extend those of Konyagin for monomials to a general polynomial. Such bounds readily yield estimates for the corresponding polynomial Waring problem mod $p$, namely the smallest $\gamma$ such that $f(x_1)+\cdots +f(x_{\gamma})\equiv N$ (mod $p$) is solvable in integers for any $N$.