  Recent progress on Vinogradov's mean value theorem has resulted in improved
estimates for exponential sums of Weyl type. We apply these new estimates to
obtain sharper bounds for the function $H(k)$ in the Waring--Goldbach problem.
We obtain new results for all exponents $k\ge 8$, and in particular establish
that $H(k)\le (4k-2)\log k+k-7$ when $k$ is large, giving the first improvement
on the classical result of Hua from the 1940s.
