Short cubic exponential sums over primes

For $y\ge x^{4/5}\mathscr L^{8B+151}$ (where $\mathscr L=\log (xq)$ and $B$ is an absolute constant), a nontrivial estimate is obtained for short cubic exponential sums over primes of the form $S_3(\alpha ;x,y) = \sum _{x-y<n\le x}\Lambda (n) e(\alpha n^3)$, where $\alpha =a/q+\theta /q^2$, $(a,q)=1$, $\mathscr L^{32(B+20)}<q\le y^5x^{-2}\mathscr L^{-32(B+20)}$, $|\theta |\le 1$, $\Lambda$ is the von Mangoldt function, and $e(t)=e^{2\pi it}$.