  The mKP equation with self-consistent sources (mKPESCS) is treated in the
framework of the constrained mKP hierarchy. We introduce a new constrained mKP
hierarchy which may be viewed as the stationary hierarchy of the mKP hierarchy
with self-consistent sources. This offers a natural way to obtain the Lax
representation for the mKPESCS. Based on the conjugate Lax pairs, we construct
the generalized Darboux transformation with arbitrary functions in time $t$ for
the mKPESCS which, in contrast with the Darboux transformation for the mKP
equation, provides a non-auto-B\"{a}cklund transformation between two mKPESCSs
with different degrees. The formula for $n$-times repeated generalized Darboux
transformation is proposed and enables us to find the rational solutions
(including the lump solutions), soliton solutions and the solutions of breather
type of the mKPESCS.
