  We analyse the production of a light charged Higgs boson at the Large Hadron
Collider (LHC) via the quark-fusion mechanism $c \bar{b} \to H^- $ considering
the decay channel $H^- \to \tau \bar \nu_\tau$ in the final state. We study
this process in the framework of the 2-Higgs Doublet Model Type III (2HDM-III)
which assumes a four-zero texture in the Yukawa matrices and a general Higgs
potential, wherein the two Higgs doublets coupling to both up and down fermions
do generate Flavour Changing Neutral Currents (FCNCs) yet the latter can be
controlled by the texture when flavour physics constraints are considered. We
consider the parameter space of the model where this signal is enhanced and in
agreement with both theoretical constraints and experimental data. In
particular, we exploit the setup with lepton-specific-like Yukawa couplings and
assess the LHC sensitivity to such $H^\pm$ signals against the dominant
{irreducible and} reducible backgrounds. We show that in our model BR$(H^\pm
\to cb) \sim 0.1- 0.2$ and BR$(H^\pm \to \tau \nu) \sim 0.7-0.9$ so that, under
these conditions, the prospects for $H^\pm$ detection in the 2HDM-III in the
aforementioned production and decay channels are excellent assuming standard
collider energy and luminosity conditions.
