A substantial fraction of systematic uncertainties in neutrino oscillation experiments stems from the lack of precision in modeling the nucleus when describing neutrino-nucleus interactions. Reducing these uncertainties is crucial for present and next-generation long baseline experiments. This can be achieved in two complementary ways: improved detectors and more precise neutrino interaction models. The T2K experiment is preparing for its second phase with the upgrade of its near detector starting next year. The capabilities of full polar angle acceptance, lower proton tracking threshold as well as reconstruction of neutron kinematics with this upgrade will open the door to explore new physics with unprecedented precision thanks to new observables. To model quasi-elastic interactions, T2K uses the Benhar Spectral Function (SF) model which offers significant improvements with respect to the more commonly used Fermi gas-based models. It features a distribution of momenta and removal energies of nucleons inside the nucleus within the shell-model picture, and also accounts for short-range correlations between nucleons. Electron scattering experiments allow a precise probe of the structure of the nucleus and have been used to both construct and validate the SF model. Based on constraints from such experiments, we develop a set of parameters that can alter the occupancy of the nuclear shells and the distribution of the nucleon momentum within each shell. In addition, the contribution of short-range correlations and the effect of Pauli blocking can also be modified. With such freedoms, uncertainties on the input models can be estimated for neutrino oscillation analyses. In this poster, we will first show how this parameterisation can improve the SF model agreement with available cross section measurements of the muon momentum and direction as well as the transverse momentum imbalance from T2K and MINERvA. Then we will present the sensitivity of the upgraded T2K near detector to these parameters for the statistics expected during the second phase of T2K and Hyper-Kamiokande.