  Recently, the CMS and ATLAS collaborations have announced the results for
$H\rightarrow Z[\rightarrow \ell^{+}\ell^{-}]\gamma$ with $\ell=e$ or $\mu$
\cite{CMS:2022ahq,CMS:2023mku}, where $H\rightarrow Z\gamma$ is a sub-process
of $H\rightarrow \ell^{+} \ell^{-} \gamma$. This semi-leptonic Higgs decay
receives loop induced resonant $H\rightarrow Z[\rightarrow
\ell^{+}\ell^{-}]\gamma$ as well as non-resonant contributions. % as discussed
in \cite{Kachanovich:2021pvx}.
  To probe further features coming from these contributions to $H\rightarrow
\ell^{+} \ell^{-} \gamma$, we argue that the polarization of the final state
leptons is also an important parameter. We show that the contribution from the
interference of resonant and non-resonant terms plays an important role when
the polarization of final state lepton is taken into account, which is
negligible in the case of unpolarized leptons. For this purpose, we have
calculated the polarized decay rates and the longitudinal ($P_L$), normal
($P_N$) and transverse ($P_T$) polarization asymmetries. We find that these
asymmetries purely come from the loop contributions and are helpful to further
investigate the resonant and non-resonant nature of $H\rightarrow Z[\rightarrow
\ell^{+}\ell^{-}]\gamma$ decay. We observe that for $\ell=e,\mu$, the
longitudinal decay rate is highly suppressed around $m_{\ell\ell}\approx 60$GeV
when the final lepton spin is $-\frac{1}{2}$, dramatically increasing the
corresponding lepton polarization asymmetries. Furthermore, we analyze another
observable, the ratio of decay rates $R^{\ell\ell'}_{i\pm}$, where $\ell$ and
$\ell'$ refer to different final state lepton generations. Precise measurements
of these observables at the HL-LHC and the planned $e^{+}e^{-}$ can provide a
fertile ground to test not only the SM but also to examine the signatures of
possible NP beyond the SM.
