We examine the impact of small parity-time (PT) symmetric perturbations on nonlinear optical honeycomb lattices in the tight-binding limit. We show for strained lattices that complex dispersion relationships do not form under perturbation, and we find a variety of nonlinear wave equations which describe the effective dynamics in this regime. The existence of semilocalized gap solitons in this case is also shown, though we numerically demonstrate these solitons are likely unstable. We show for unstrained lattices under the effect of a restricted class of PT perturbations, which prevent complex dispersion relationships from appearing, that nontrivial phase dynamics emerge as a result of the PT perturbation. This phase can be understood as momentum imparted to optical beams by the lattice, thus showing PT perturbations offer potentially novel means for the control of light in honeycomb lattices.