  The paper deals with the construction of the asymptotic soliton-like and the
asymptotic peakon-like solutions to the modified Camassa-Holm equation with
variable coefficicents and a singular perturbation. This equation is a
generalization of the well known modified Camassa-Holm equation which is
integrable system and in addition to the soliton solutions the equation has the
peakon solutions. The novelty of the ideas of this paper lies in the
development of a technique for constructing asymptotic peakon-like solutions.
In the paper a general scheme of finding asymptotic approximation of any order
is presented and accuracy of the asymptotic approximation is found.
  The obtained results are illustrated by examples both the soliton-like and
the peakon-like solutions. For the examples the equations for the phase
function as well as the main and the first terms of the soliton-like and
peakon-like solutions are found. Moreover, for different values of a small
parameter the graphs that demonstrate kind of the solutions are presented. The
considered examples demonstrate that for an adequate description of the wave
process it is enough obtain the main and the first terms of correspond
asymptotic solutions.
  The results also confirm that the proposed technique can be used for
constructing asymptotic wave-like solutions of other equations.
