  This article extends a strong averaging principle for L\'evy diffusions which
live on the leaves of a foliated manifold subject to small transversal L\'evy
type perturbation to the case of non-compact leaves. The main result states
that the existence of $p$-th moments of the foliated L\'evy diffusion for
$p\geq 2$ and an ergodic convergence of its coefficients in $L^p$ implies the
strong $L^p$ convergence of the fast perturbed motion on the time scale
$t/\epsilon$ to the system driven by the averaged coefficients. In order to
compensate the non-compactness of the leaves we use an estimate of the
dynamical system for each of the increments of the canonical Marcus equation
derived in da Costa and Hoegele (2017), the boundedness of the coefficients in
$L^p$ and a nonlinear Gronwall-Bihari type estimate. The price for the
non-compactness are slower rates of convergence, given as $p$-dependent powers
of $\epsilon$ strictly smaller than $1/4$.
