  We consider the Peccei-Quinn (PQ) mechanism as the one behind the Dirac
neutrino masses when these are generated through the $d=5$ effective operator
$\bar{L}\tilde{H}N_R\phi$ at one loop level, with $\phi$ being a Standard Model
singlet scalar. In this setup, the PQ symmetry guarantees that the one-loop
realization of such an effective operator gives the leading contribution to the
Dirac neutrino masses by forbidding the contributions arising from its tree
level realizations. All the mediators in the one-loop neutrino mass diagrams
can be stabilized by a remnant $Z_N$ symmetry from the PQ symmetry breaking,
thus forming a dark sector besides the axion sector and leading to mixed
axion/WIMP dark matter scenarios.
