  We study the average $p-$Wasserstein distance between a finite sample of an
infinite hyperuniform point process on $\mathbb{R}^2$ and its mean for any
$p\geq 1$. The average Wasserstein transport cost is shown to be bounded from
above and from below by some multiples of the number of points. More generally,
we give a control on the $p-$Wasserstein distance in function of a control on
the $L^p$ norm of the difference of the point process and its mean. We also
obtain the $d$-dimensional version of this result.
