  We establish new explicit bounds on the Gaussian approximation of Poisson
functionals based on novel estimates of moments of Skorohod integrals.
Combining these with the Malliavin-Stein method, we derive bounds in the
Wasserstein and Kolmogorov distances whose application requires minimal moment
assumptions on add-one cost operators $\unicode{x2014}$ thereby extending the
results from (Last, Peccati and Schulte, 2016). Our applications include a CLT
for the Online Nearest Neighbour graph, whose validity was conjectured in
(Wade, 2009; Penrose and Wade, 2009). We also apply our techniques to derive
quantitative CLTs for edge functionals of the Gilbert graph, of the $k$-Nearest
Neighbour graph and of the Radial Spanning Tree, both in cases where
qualitative CLTs are known and unknown.
