Let \(\Lambda\) be the von Mangoldt function, \(\mu\) be the Möbius function, \(\theta\) be a real larger than \(2/3\), and \(g\) be a polynomial of degree \(k\geq 1\). In the paper under review, the authors obtain estimate of the form \(H/(\log N)^A\) for the sums \[ \sum_{N<n\leq N+N^\theta}\Lambda(n)\,e^{2\pi i g(n)},\qquad \sum_{N<n\leq N+N^\theta}\mu(n)\,e^{2\pi i g(n)}. \] The first sum in the special case \(g(n)=\alpha n^k\) implies a short interval version of the Waring-Goldbach problem, regarding to writing \(N\) as the sum of \(k\)th power of primes.