With the deeper study of Higgs particle, Higgs precision measurements can be served to probe new physics indirectly. In many new physics models, vector-like quarks $T_L,~T_R$ occur naturally. It is important to probe their couplings with standard model particles. In this work, we consider the singlet $T_L,~T_R$ extended models and show how to constrain the $Tth$ couplings through the $hrightarrowgamma Z$ decay at high-luminosity LHC. Firstly, we derive the perturbative unitarity bounds on $|y_{L,~R}^{tT}|$ with other couplings set to be zeros simply. To optimize the situation, we take $m_T$ = 400 GeV and $s_L$ = 0.2 considering the experimental constraints. Under this benchmark point, we find that the future bounds from $hrightarrowgamma Z$ decay can limit the real parts of $y_{L,~R}^{tT}$ in the positive direction to be O(1) because of the double enhancement. For the real parts of $y_{L,~R}^{tT}$ in the negative direction, it is always surpassed by the perturbative unitarity. Moreover, we find that the top quark electric dipole moment can give stronger bounds (especially the imaginary parts of $y_{L,~R}^{tT}$) than the perturbative unitarity and $hrightarrowgamma Z$ decay in the off-axis regions for some scenarios.
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