  We study the production of a neutral Higgs boson at a Future Circular
Collider in the electron-hadron mode (FCC-eh) through the leading process $e^-
p \to \nu_e h q $ assuming the decay channel $h \to c \bar{c} $, where $h$ is
the Standard Model (SM)-like state discovered at the Large Hadron Collider
(LHC). This process is studied in the context of a 2-Higgs Doublet Model Type
III (2HDM-III) embedding a four-zero texture in the Yukawa matrices and a
general Higgs potential, where both Higgs doublets are coupled with up- and
down-type fermions. Flavour Changing Neutral Currents (FCNCs) are well
controlled by this approach through the adoption of a suitable texture once
flavour physics constraints are taken in account. Considering the parameter
space where the signal is enhanced and in agreement with both experimental data
and theoretical conditions, we analyse the aforementioned signal by taking into
account the most important SM backgrounds, separating $c$-jets from
light-flavour and gluon ones as well as $b$-jets by means of efficient flavour
tagging. We find that the $h c \bar{c} $ coupling strength can be accessed with
good significance after a luminosity of $1$ ab$^{-1}$ for a 50 TeV proton beam
and a 60 GeV electron one, the latter with a 80\% (longitudinal) polarisation.
