  We prove two bounds for discrete moments of Weyl sums. The first one can be
obtained using a standard approach. The second one involves an observation how
this method can be improved, which leads to a sharper bound in certain ranges.
The proofs both build on the recently proved main conjecture for Vinogradov's
mean value theorem.
  We present two selected applications: First, we prove a new $k$- th
derivative test for the number of integer points close to a curve by an
exponential sum approach. This yields a stronger bound than existing results
obtained via geometric methods, but it is only applicable for specific
functions. As second application we prove a new improvement of the polynomial
large sieve inequality for one-variable polynomials of degree $k \geq 4$.
