  Considering the framework of the 2-Higgs Doublet Model Type III (2HDM-III),
wherein two doublets are coupled both to up as down fermions, and the Flavour
Changin Currents (FCNCs) is controlled by a four-zero Texture approach in the
Yukawa matrices. We study the production of charged Higgs ($H^\pm$) by means of
$cb$ quark fusion followed by channel decay $H^\pm \to \tau^\pm \nu$ : $ c b
\to H^\pm \to \tau^\pm \bar \nu $. Taking in account the experimental bounds,
as well as the theoretical constraints analysed in previous work
arXiv:2003.06263[hep-ph]. One can show that in a lepton-specific-like
incarnation of the model, we obtain a significant sensitivity for the LHC
machine. We consider in the aforementioned study to both irreducible and
reducible backgrounds of the SM.
