Discorrelation between primes in short intervals and polynomial phases

Let $H=N^{\theta },\theta >2\mathrm{/}3$ and $k\ge 1$. We obtain estimates for the following exponential sum over primes in short intervals: $$\sum _{N<n\le N+H}\mathrm{\Lambda }(n)e(g(n)),$$ 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.