  We studied the one-loop prediction for the single production of a Higgs-like
boson in association with a photon in electron-positron collisions in the
context of two Higgs doublet type-II seesaw model (2HDMcT). We explored to what
extent the new scalars in the (2HDMcT) spectrum affect its production
cross-section, the ratio $R_{\gamma h_1}$ as well as the signal strengths
$R_{\gamma Z}$ and $R_{\gamma \gamma}$ when $h_1$ is identified with the
observed $SM $ Higgs boson within the $2HDMcT$ delimited parameter space. More
specifically, we focused on $e^+e^- \to h_1\gamma$ process at one-loop, and
analyzed how it evolves under a full set of theoretical constraints and the
available experimental data, including $B\to X_s\gamma$ constraint at 95$\%$
C.L. Our analysis showed that these observables are particularly sensitive to
the parameters $\alpha_1$, $\lambda_{7}$, $\lambda_{9}$, the trilinear Higgs
couplings and to the charged Higgs masses. We found that $\sigma (e^+e^- \to
h_1\gamma)$ can significantly be enhanced up to $8.1\,\, 10^{-2}$ fb, so
exceeding the Standard Model prediction. Additionally, as a byproduct, we also
found that $R_{\gamma h_1}$ is entirely correlated with both the
$h_1\to\gamma\gamma$ and $h_1\to\gamma Z$ signal strengths.
