  In this paper, we present a dispersive regularization for the modified
Camassa-Holm equation (mCH) in one dimension, which is achieved through a
double mollification for the system of ODEs describing trajectories of
$N$-peakon solutions. From this regularized system of ODEs, we obtain
approximated $N$-peakon solutions with no collision between peakons. Then, a
global $N$-peakon solution for the mCH equation is obtained, whose trajectories
are global Lipschitz functions and do not cross each other. When $N=2$, the
limiting solution is a sticky peakon weak solution. By a limiting process, we
also derive a system of ODEs to describe $N$-peakon solutions. At last, using
the $N$-peakon solutions and through a mean field limit process, we obtain
global weak solutions for general initial data $m_0$ in Radon measure space.
