Charged-current cross sections are calculated for quasielastic neutrino and antineutrino scattering using a relativistic meson-nucleon model. We examine how nuclear-structure effects, such as relativistic random-phase-approximation (RPA) corrections and momentum-dependent nucleon self-energies, influence the extraction of the axial form factor of the nucleon. RPA corrections are important only at low-momentum transfers. In contrast, the momentum dependence of the relativistic self-energies changes appreciably the value of the axial-mass parameter MA extracted from dipole fits to the axial form factor. Using Brookhaven's experimental neutrino spectrum we estimate the sensitivity of MA to various relativistic nuclear-structure effects.