  We establish inequalities for assessing the distance between the distribution
of a (possibly multidimensional) functional of a Poisson random measure and
that of a Gaussian element. Our bounds only involve add-one cost operators at
the order one - that we evaluate and compare at two different scales - and are
specifically tailored for studying the Gaussian fluctuations of sequences of
geometric functionals displaying a form of weak stabilization - see Penrose and
Yukich (2001) and Penrose (2005). Our main bounds extend the estimates recently
exploited by Chatterjee and Sen (2017) in the proof of a quantitative version
of the central limit theorem (CLT) for the length of the Poisson-based
Euclidean minimal spanning tree (MST). We develop in full detail three
applications of our bounds, namely: (i) to a quantitative multidimensional
spatial CLT for functionals of the on-line nearest neighbor graph, (ii) to a
quantitative multidimensional CLT involving functionals of the empirical
measure associated with the edge-length of the Euclidean MST, and (iii) to a
collection of multidimensional CLTs for geometric functionals of the excursion
set of heavy-tailed shot noise random fields. Application (i) is based on a
collection of general probabilistic approximations for strongly stabilizing
functionals, that is of independent interest.
