Author’s abstract: We prove the following estimate on exponential sums over primes: Let \(k \geq 1\), \(\beta_k = 1/2 + \log k / \log 2\), \(x \geq 2\) and \(\alpha = a/q + \lambda\) subject to \((a,q)=1\), \(1 \leq a \leq q\) and \(\lambda \in \mathbb{R}\) . Then \[ \sum_{x < m \leq 2x} \Lambda(m) e(\alpha m^k) ~\ll~ (d(q))^{\beta_k} (\log x)^c \left( x^{1/2} \sqrt{q (1+ | \lambda | x^k)} + x^{4/5} + \frac{x}{\sqrt{q(1+| \lambda| x^k)}} \right). \] As an application, we prove that with at most \(O(n^{7/8 + \varepsilon})\) exceptions, all positive integers up to \(N\) satisfying some necessary congruence conditions are the sum of three squares of primes. This result is as strong as what has previously been established under the generalized Riemann hypothesis.