  In this paper, we study traveling wave solutions and peakon weak solutions of
the modified Camassa-Holm (mCH) equation with dispersive term $2ku_x$ for
$k\in\mathbb{R}$. We study traveling wave solutions through a Hamiltonian
system obtained from the mCH equation by using a nonlinear transformation. The
typical traveling wave solutions given by this Hamiltonian system are unbounded
or multi-valued. We provide a method, called patching technic, to truncate
these traveling wave solutions and patch different segments to obtain patched
bounded single-valued peakon weak solutions which satisfy jump conditions at
peakons. Then, we study some special peakon weak solutions constructed by the
fundamental solution of the Helmholtz operator $1-\partial_{xx}$, which can
also be obtained by the patching technic. At last, we study some length and
total signed area preserving closed planar curve flows that can be described by
the mCH equation when $k=1$, for which we give a Hamiltonian structure and use
the patched periodic peakon weak solutions to investigate loops with cusps.
