  TO BE PUBLISHED BY ISRAEL JOURNAL OF MATHEMATICS.
  We study the mean $\sum_{x\in\mathcal{X}} \bigl|\sum_{p\le N}{}u_p
e(xp)\bigr|^{\ell}$ when $\ell$ covers the full range $[2,\infty)$ and
$\mathcal{X}\subset\mathbb{R}/\mathbb{Z}$ is a {well-spaced} set, providing a
smooth transition from the case $\ell=2$ to the case $\ell>2$ and improving on
the results of J.~Bourgain and of B.~Green and T.~Tao. A uniform
Hardy-Littlewood property for the set of primes is established as well as a
sharp upper bound for $\sum_{x\in\mathcal{X}} \bigl|\sum_{p\le N}{}u_p
e(xp)\bigr|^{\ell}$ when $\mathcal{X}$ is small. These results are extended to
primes in \emph{any} interval in a last section, provided the primes are
numerous enough therein.
