Given an NQC log canonical generalized pair (X, B + M) whose underlying variety X is not necessarily Q-factorial, we show that one may run a (KX + B + M)-MMP with scaling of an ample divisor which terminates, provided that (X, B + M) has a minimal model in a weaker sense or that KX + B + M is not pseudo-effective. We also prove the existence of minimal models of pseudo-effective NQC log canonical generalized pairs under various additional assumptions, for instance when the boundary contains an ample divisor.