  Precise knowledge of the nucleon's axial-current form factors is crucial for
modeling GeV-scale neutrino-nucleus interactions. Unfortunately, the axial form
factor remains insufficiently constrained to meet the precision requirements of
upcoming long-baseline neutrino-oscillation experiments. This work studies the
nucleon's axial and vector form factors using the light-front approach to build
a quark-diquark model of the nucleon with an explicit pion cloud. The
light-front wave functions in both the quark and pion-baryon Fock spaces are
first calibrated to existing experimental information on the nucleon's
electromagnetic form factors, and then used to predict the axial form factor.
The resulting squared charge radius of the axial pseudo-vector form factor is
predicted to be $r_A^2\! =\! 0.29\! \pm\! 0.03\, \mathrm{fm}^2$, where the
small error accounts for the model's parametric uncertainty. We use our form
factor results to explore the (quasi-)elastic scattering of neutrinos by
(nuclei)nucleons, with the result that the the widely-implemented dipole ansatz
is an inadequate approximation of the full form factor for modeling both
processes. The approximation leads to a $5\!-\!10\%$ over-estimation of the
total cross section, depending on the (anti)neutrino energy. We project
over-estimations of similar size in the flux-averaged cross sections for the
upcoming DUNE long-baseline neutrino-oscillation experiment.
