  We study models that produce a Higgs boson plus photon ($h^0 \gamma$)
resonance at the LHC. When the resonance is a $Z'$ boson, decays to $h^0
\gamma$ occur at one loop. If the $Z'$ boson couples at tree-level to quarks,
then the $h^0 \gamma$ branching fraction is typically of order $10^{-5}$ or
smaller. Nevertheless, there are models that would allow the observation of $Z'
\to h^0 \gamma$ at $\sqrt{s} = 13$ TeV with a cross section times branching
fraction larger than 1 fb for a $Z'$ mass in the 200--450 GeV range, and larger
than 0.1 fb for a mass up to 800 GeV. The 1-loop decay of the $Z'$ into lepton
pairs competes with $h^0 \gamma$, even if the $Z'$ couplings to leptons vanish
at tree level. We also present a model in which a $Z'$ boson decays into a
Higgs boson and a pair of collimated photons, mimicking an $h^0 \gamma$
resonance. In this model, the $h^0 \gamma$ resonance search would be the
discovery mode for a $Z'$ as heavy as 2 TeV. When the resonance is a scalar,
although decay to $h^0 \gamma$ is forbidden by angular momentum conservation,
the $h^0$ plus collimated photons channel is allowed. We comment on prospects
of observing an $h^0 \gamma$ resonance through different Higgs decays, on
constraints from related searches, and on models where $h^0$ is replaced by a
nonstandard Higgs boson.
