What are the best bounds currently known for the following exponential sum:
$$\sum_{x < p \le 2x} e(\alpha p^k)$$
for values of $\alpha$ far from a rational with small denominator. ($p$ refers to a prime in this case)
In particular, I am interested in such bounds for $k = 9$.
The following gives some bounds: https://projecteuclid.org/download/pdf_1/euclid.mmj/1156345592, but have these bounds been improved since then?
EDIT: I am also interested in such bounds assuming GRH.