  We investigate the possibility of probing the CP-violating $h\bar{t}t$
coupling in the process $e^{+}e^{-}\rightarrow h \gamma$ at the future high
luminosity $e^{+}e^{-}$ colliders. Our numerical results show that the cross
section for this process can be significantly increased for the allowed CP
phase $\xi$ and center of mass energy. For example the cross section is about 6
times of that in the standard model (SM) for $\sqrt{s}=350~\mathrm{GeV}$ and
$\xi =3\pi/5$ (see text for $\xi$ definition). The simulation for the signal
process $e^{+}e^{-}\rightarrow h \gamma \rightarrow b \bar{b}\gamma$ and its
backgrounds shows that the signal significance can reach more than $4.8\sigma$
and $2.3\sigma$ for $\sqrt{s}=350~\mathrm{GeV},\ 500~\mathrm{GeV}$
respectively, with the integrated luminosity $\mathcal{L}=3~\text{ab}^{-1}$ and
$\xi\in[\pi/2,3\pi/5]$. For $\mathcal{L}=10~\text{ab}^{-1}$, the signal
significance can be greater than 5 for $\sqrt{s}=350~\mathrm{GeV}$ and 3.9 for
$\sqrt{s}=500~\mathrm{GeV}$ with the CP phase $\xi\in[\pi/2,3\pi/5]$. Besides
the cross section enhancement, the CP-violating $h\bar{t}t$ coupling will
induce the forward-backward asymmetry $A_{FB}$ which is absent in the SM and is
a clear signal of new CP violation. Compared with the $A_{FB}$ in the Higgs
decay $h\rightarrow l^{+}l^{-}\gamma$, the $A_{FB}$ can be greatly enhanced in
the production process. For example $A_{FB}$ can reach 0.5 for $\cos\xi\simeq
0.7$ and $\sqrt{s}=500~\mathrm{GeV}$. Due to the large backgrounds, the
significance of the expected $A_{FB}$ can only be observed at $1.7\sigma$ with
$\mathcal{L}=10~\text{ab}^{-1}$ and $\sqrt{s}=500~\mathrm{GeV}$. It is
essential to trigger the single photon in the final state to separate the
bottom jets arising from scalar or vector bosons, in order to isolate the
signal from the backgrounds more efficiently.
