  In this paper, we explore the signature of a simplified model which includes
a new singlet scalar state and a vector like quark at future lepton colliders.
In particular, we study the production of the new singlet scalar in association
with a photon, which proceeds through loop level diagrams involving vector like
top quark partner, at future $e^-e^+$ colliders with different center-of-mass
energies from 500 GeV to 3 TeV. To show the sensitivity of the process, the
exclusion limits on the parameter space of the model are presented considering
the decay of the singlet scalar into a pair of Higgs boson, followed by the
decay of Higgs bosons into $b\bar{b}$ pairs. The results are compared to those
obtained from the LHC, electroweak precision data and other channels at lepton
colliders and it is shown that a notable sensitivity to the parameter space of
the simplified model could be achieved.
