  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 charge 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.
