We study the production and decay of a light charged Higgs boson at the future Large Hadron electron Collider (LHeC) in the framework of the Two Higgs Doublet Type III, assuming a four-zero texture in the Yukawa matrices and a general Higgs potential. We analyze the charged current production processes $e^- p \to \nu_e q H^+ $ considering the signature $H^+ \to c \bar{b} + c.c.$ of the final state. We take this signature and we compare it to the irreducible background from Standard Model (SM) interactions. We consider scenarios of the model which are consistent with current experimental data from Higgs and flavor physics.