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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  ⩾  8   , and in particular establish that    H  (  k  )  ⩽  (  4  k  -  2  )  log  k  +  k  -  7    when   k   is large, giving the first improvement on the classical result of Hua from the 1940s.

On the Waring–Goldbach problem for eighth and higher powers