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Edit: I made a mistake, my attempted answer is wrong. It was just the same as @Felipe's comment, namely, that Jason Starr's answer to this question might apply here too. But it doesn't. Sorry for the mistake.

Edit: I made a mistake, my attempted answer is wrong. It was just the same as @Felipe's comment, namely, that Jason Starr's answer to this question might apply here too. But it doesn't. Sorry for the mistake.

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.

Edit: I made a mistake, my attempted answer is wrong. It was just the same as @Felipe's comment, namely, that Jason Starr's answer to this question might apply here too. But it doesn't. Sorry for the mistake.

The answer is "no", for the same reason that Jason Starr explained in his answer to mathoverflow:118117.

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.