Discorrelation between primes in short intervals and polynomial phases

Let $H = N^{\theta}, \theta > 2/3$ and $k \geq 1$. We obtain estimates for the following exponential sum over primes in short intervals: \[ \sum_{N < n \leq N+H} \Lambda(n) e(g(n)), \] where $g$ is a polynomial of degree $k$. As a consequence of this in the special case $g(n) = \alpha n^k$, we deduce a short interval version of the Waring-Goldbach problem.