  In this work we improve on several aspects of the computation of the
(anti-)neutrino charged-current multi-nucleon cross section carried out in
Phys.Rev.C 83 (2011) 045501 and Phys.Rev.C 102 (2020) 024601. Most importantly,
we implement a consistent treatment of the nucleon self-energy in the $W^\pm
N\to N'\pi$ amplitude entering the definition of the two-particle two-hole
(2p2h) cross-section, and estimate the source of uncertainty of our model due
to a simplified treatment of the $\Delta$ self-energy. Our new predictions are
around $20-40\%$ higher than previously. We show comparisons for the inclusive
lepton double-differential cross sections, with no pions in the final state,
measured by MiniBooNE on carbon and by T2K on carbon and oxygen. In all cases,
we find an excellent reproduction of the experiments, and in particular, the
neutrino MiniBooNE data is now well described without requiring a global $90\%$
re-scaling of the flux. In addition, we take the opportunity of this revision
to discuss in detail several important issues of the calculation of the 2p2h
cross section, delving into the microscopic dynamics of the multi-nucleon
mechanisms. The improved treatment presented in this work provides realistic
first-step emitted two-nucleon final state momentum configurations, beyond the
approximation of phase-space distributions.
