Estimates of short exponential sums with primes in major arcs

For a number of additive problems with almost equal summands, in addition to the estimates for short exponential sums with primes of the form
$$ S_k(\alpha;x,y)=\sum_{x-y<n\le x}\Lambda(n)e(\alpha n^k), $$
in minor arcs, we need to have an estimate of these sums in major arcs, except for a small neighborhood of their centers. We also need to have an asymptotic formula on a small neighborhood of the centers of major arcs.
In this paper, using the second moment of Dirichlet $L$-functions on the critical line, we obtained a nontrivial estimate of the form
$$ S_k(\alpha;x,y)\ll y\mathscr{L}^{-A}, $$
for $S_k(\alpha;x,y)$ in major arcs $M(\mathscr{L}^b)$, $\tau=y^5x^{-2}\mathscr{L}^{-b_1}$, $\mathscr{L} =\ln xq$, except for a small neighborhood of their centers $|\alpha-\frac{a}{q}|>\left(2\pi k^2x^{k-2}y^2\right)^{-1}$, when $y\ge x^{1-\frac{1}{2k-1+\eta_k}}\mathscr{L}^{c_k}$, where
$$ \eta_k=\frac{2}{4k-5+2\sqrt{(2k-2)(2k-3)}}, c_k= \frac{2A+22+\left(\frac{2\sqrt{2k-3}}{\sqrt{2k-2}}-1\right)b_1}{2\sqrt{(2k-2)(2k-3)}-(2k-3)}, $$
and $A$, $b_1$, $b$ are arbitrary fixed positive numbers. Furthermore, and we also proved an asymptotic formula on a small neighborhood of the centers of major arcs.