  In the traditional neutrinophilic two-Higgs doublet model ($\nu$2HDM), there
is no non-standard neutrino interaction (NSI) as the interactions between the
Standard Model fermions with neutrinos are negligibly small due to the tiny
mixing of the two scalar doublets. In this work, we propose that if $\nu$2HDM
is modified by considering the right-handed electron, $e_R$ is negatively
charged under a global $U(1)$-symmetry then one can generate significant amount
of NSI along with the tiny Dirac neutrino mass. Depending on different
constraints from the LEP experiment, tree level lepton flavor violating
processes, big-bang neucleosynthesis etc., we observe that this model
significantly restricts the range of permissible NSI parameters, putting a
strict upper bound on different NSIs. Furthermore, considering these
model-dependent NSIs, we study their impact on the next-generation superbeam
experiment, DUNE. We present a detailed discussion on the mass hierarchy
sensitivity and the CP-violation discovery study considering the impact of both
diagonal as well as off-diagonal NSIs.
