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The

Edit: I made a mistake, my attempted answer is "no", for wrong. It was just the same reason as @Felipe's comment, namely, that Jason Starr explained in his Starr's 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 pointthis question might apply here too. 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 it doesn't. Sorry for the deformation space is a rational varietymistake.

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.