  We investigate the sensitivity of top-Higgs coupling by considering the
associated vertex as CP phase ($\zeta_t$) dependent through the process $p\,
e^- \to \bar t \,h \,\nu_e$ in the future Large Hadron electron collider. In
particular the decay modes are taken to be $h \to b\bar b$ and $\bar t \to$
leptonic mode. Several distinct $\zeta_t$ dependent features are demonstrated
by considering observables like cross sections, top-quark polarisation,
rapidity difference between $h$ and $\bar t$ and different angular asymmetries.
Luminosity ($L$) dependent exclusion limits are obtained for $\zeta_t$ by
considering significance based on fiducial cross sections at different
$\sigma$-levels. For electron and proton beam-energies of 60 GeV and 7 TeV
respectively, at $L = 100$~fb$^{-1}$, the regions above $\pi/5 < \zeta_t \leq
\pi$ are excluded at 2$\sigma$ confidence level, which reflects better
sensitivity expected at the Large Hadron Collider. With appropriate error
fitting methodology we find that the accuracy of SM top-Higgs coupling could be
measured to be $\kappa = 1.00 \pm 0.17 (0.08)$ at $\sqrt{s} = 1.3 (1.8)$ TeV
for an ultimate $L = 1 \,\rm{ab}^{-1}$.
