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.