  We study the impact of nuclear structure uncertainties on a measurement of
the weak charge of $^{12}\mathrm{C}$ at the future MESA facility in Mainz.
Information from a large variety of nuclear models, accurately calibrated to
the ground-state properties of selected nuclei, suggest that a $0.3$% precision
measurement of the parity-violating asymmetry at forward angles will not be
compromised by nuclear structure effects, thereby allowing a world-leading
determination of the weak charge of $^{12}\mathrm{C}$. Further, we show that a
combination of measurements of the parity-violating asymmetry at forward and
backward angles for the same electron beam energy can be used to extract
information on the nuclear weak charge distribution. We conclude that a $0.34$%
precision on the weak radius of $^{12}\mathrm{C}$ may be achieved by performing
a $3$% precision measurement of the parity-violating asymmetry at backward
angles.
