  In this paper we present the first study of the impact of the
$\mathcal{O}(\alpha)$ EW correction to the $pp \to ZZ \gamma+X$ process at the
CERN Large Hadron Collider (LHC). The subsequent $Z$-boson leptonic decays are
considered at the leading order using the MadSpin method, which takes into
account the spin-correlation and off-shell effects from the $Z$-boson decays.
We provide numerical results of the integrated cross section and the kinematic
distributions for this process. In coping with final-state photon-jet
separation in the QCD real emission and photon-induced processes, we adopt both
the Frixione isolated-photon plus jets algorithm and the phenomenological
quark-to-photon fragmentation function method for comparison. We find that the
next-to-leading order (NLO) EW correction to the $ZZ\gamma$ production can be
sizeable and amounts to about $-7\%$ of the integrated cross section, and
provides a non$-$negligible contribution to the kinematic distributions,
particularly in the high energy region. We conclude that the NLO EW correction
should be included in precision theoretical predictions in order to match
future experimental accuracy.
