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We perform an updated analysis on the searches for the anomalous FCNC Yukawainteractions between the top quark, the Higgs boson, and either an up or charmquark ($\rm tqh,\ q=u,\ c$). We probe the observability of the FCNC top-Higgscouplings through the processes $\rm e^- p\rightarrow \nu_e \bar{t} \rightarrow\nu_e h \bar{q}$ (signal.I) and $\rm \ e^- p \to \nu_e h b$ (singal.II) at theproposed electron proton (ep) colliders, where the Higgs boson decays to a $\rmb\bar{b}$ pair. We find that at the high luminosity (1 $\rm ab^{-1}$) epcolliders where the electrons have a polarisation of $\rm 80\%$ and electronenergy is typical 60 GeV, the 2$\sigma$ upper limit on $\rm Br(t\to uh)$ are$0.15\times 10^{-2}$($2.9\times 10^{-4}$) at the 7TeV@LHeC(50TeV@FCC-eh) forsignal.I and $0.15\times 10^{-2}$($2.2\times 10^{-4}$) for signal.II. We alsogive an estimate on how the sensitivity (take signal.I as examples) wouldchange when we reduce the electron beam energy from 60 GeV to 50 GeV or even 40GeV due to the cost reason. The conclusion is that the discovery potentialreduce $8.7\%$($29.4\%$) if the electron beam change from 60GeV to 50(40) GeVat the 7TeV LHeC, and $16.8\%$($19.8\%$) at the 50 TeV FCC-eh.

Exploring the Anomalous Top-Higgs FCNC Couplings at the electron proton colliders