The answer is "no", for the same reason that Jason Starr explained in his answer to mathoverflow:118117.
Edit: in more detail, the point is that for $g\ge 23$, there is no nonconstant morphism from a rational variety to $M_g$ whose image contains the general point. But if a general genus-$g$ curve $C$ admitted a morphism $C'\to C$ from a smooth plane curve $C'$, then we could deform $C'$ and the morphism so that the image still has genus $g$. This gives a contradiction, since the deformation space is a rational variety.