Recent progress on Vinogradov’s mean value theorem obtained by the second author [Ann. Math. (2) 175, No. 3, 1575–1627 (2012; Zbl 1267.11105); Adv. Math. 294, 532–561 (2016; Zbl 1365.11097); Proc. Lond. Math. Soc. (3) 111, No. 3, 519–560 (2015; Zbl 1328.11087)] permits improved estimates for exponential sums of Weyl type. The authors apply these new estimates to obtain sharper bounds for the function \(H(k)\) in the Waring-Goldbach problem. For an integer \(k\geq 2\) and a positive prime number \(p\) with \(p^\theta\mid k\), \(p^{\theta+1}\nmid k\), let \[ \gamma=\gamma(k,p)= \begin{cases} \theta+2\;\text{when }p=2\;\text{and}\;\theta>0,\\ \theta+1\;\text{otherwise}\end{cases}\qquad\text{and }K(k)= \prod_{(p-1)\mid k} p^\gamma. \] Let \(H(k)\) denote the least integer \(s\) such that every sufficiently large positive integer congruent to \(s\) modulo \(K(k)\) may be written as the sum of the \(k\)th powers of \(s\) primes. The authors obtain new results for all exponents \(k\geq 8\), and in particular establish that \(H(k)\leq (4k-2)\log k+k-7\) when \(k\) is large, giving the first improvement on the classical result of L.-K. Hua [Q. J. Math., Oxf. Ser. 9, 68–80 (1938; Zbl 0018.29404)].