  We study the $ZH$ associated production followed by the Higgs $H\to \gamma
\bar{\gamma}$ decay into a photon plus an invisible and massless dark photon,
at future high-energy $e^+e^-$ facilities. Large $H\to \gamma \bar{\gamma}$
decay rates (with branching ratios up to a few percent) are allowed, thanks to
possible non-decoupling properties of the Higgs boson under specific
conditions, and unsuppressed dark-photon couplings in the dark sector. Such
large decay rates can be obtained in the framework of recent flavor models that
aim to naturally explain the observed spread in the fermion mass spectrum. We
analyze the experimental prospects for observing the $e^+e^-\rightarrow ZH$
process followed by the semi invisible Higgs decay into a photon plus a
massless invisible system. Search strategies for both the leptonic and the
hadronic final states (arising from $Z\rightarrow \mu^+\mu^-$ and $Z\rightarrow
q\bar{q}$, respectively) are outlined. We find that a $5\sigma$ sensitivity to
a branching fraction $BR_{\gamma\bar{\gamma}}\sim 3\times 10^{-4}$ can be
achieved by combining the two channels with an integrated luminosity of 10
ab$^{-1}$ at a c.m. energy of 240 GeV. This is considerably better than the
corresponding sensitivity in alternative channels previously studied at lepton
colliders. The analysis is model independent, and its results can be
straightforwardly applied to the search of any Higgs two-body decay into a
photon plus an undetected light particle.
