This article shows a strong averaging principle for diffusions driven by discontinuous heavy-tailed Levy noise, which are invariant on the compact horizontal leaves of a foliated manifold subject to small transversal random perturbations. We extend a result for such diffusions with exponential moments and bounded, deterministic perturbations to diffusions with polynomial moments of order \(p\geqslant 2\), perturbed by deterministic and stochastic integrals with unbounded coefficients and polynomial moments. The main argument relies on a result of the dynamical system for each individual jump increments of the corresponding canonical Marcus equation. The example of Levy rotations on the unit circle subject to perturbations by a planar Levy-Ornstein-Uhlenbeck process is carried out in detail.