At leading order in weak and electromagnetic couplings, cross sections for (anti)neutrino-nucleon elastic scattering are determined by vector and axial-vector form factors. Radiative corrections in the Standard Model, and potential new physics contributions beyond the Standard Model, generate additional operators with corresponding invariant amplitudes. We provide the definition of these amplitudes and study constraints from existing experimental data. We explore the impact of modern and future cross-section measurements, considering both unpolarized and polarized observables, on constraining these amplitudes and discuss the effects of radiative corrections on the observables of interest.