  Pion production on nuclei constitutes a significant part of the total cross
section in experiments involving few-GeV neutrinos. Combined analyses of data
on deuterium and heavier nuclei points to tensions between the bubble chamber
data and the data of the MINER$\nu$A experiment, which are often ascribed to
unspecified nuclear effects. To understand the origin of these tensions, a
microscopic quantum mechanical framework is needed to compute nuclear matrix
elements. We use the local approximation to the relativistic distorted wave
impulse approximation (RDWIA) to assess the role of final-state nucleon
distortion. To perform this comparison under conditions relevant to neutrino
experiments, we compute cross sections for the MINER$\nu$A and T2K charged pion
production datasets. The inclusion of nucleon distortion leads to a reduction
of the cross section up to 10\%, but to no significant change in shape of the
flux-averaged cross sections. Results with and without distortion compare
favorably to experimental data, with the exception of the low-$Q^2$ MINER$\nu$A
$\pi^+$ data. We point out that hydrogen target data from BEBC is also
overpredicted at low-$Q^2$, and that the discrepancy is similar in shape and
magnitude to what is found in comparison to MINER$\nu$A data. Including nucleon
distortion alone cannot explain the overprediction of low-$Q^2$ cross sections
measured by MINER$\nu$A. The similar overprediction of BEBC data on hydrogen
means that it is impossible to ascribe this discrepancy solely to a nuclear
effect. Axial couplings and their $Q^2$ dependence should ideally be derived
from more precise data on hydrogen and deuterium.
