  We consider the MiNNLO$_{\rm PS}$ method to consistently combine
next-to-next-to-leading order (NNLO) QCD calculations with parton-shower
simulations. We identify the main sources of differences between MiNNLO$_{\rm
PS}$ and fixed-order NNLO predictions for inclusive observables due to
corrections beyond NNLO accuracy and present simple prescriptions to either
reduce or remove them. Refined predictions are presented for Higgs, charged-
and neutral-current Drell Yan production. The agreement with fixed-order NNLO
calculations is considerably improved for inclusive observables and scale
uncertainties are reduced. The codes are released within the POWHEG-BOX.
