We present one-loop contributing
for $h\rightarrow \ell \bar{\ell}\gamma$
with $\ell =\nu_{e,\mu, \tau}, e, \mu$ 
and $e^-e^+\rightarrow h\gamma$ 
in $U(1)_{B-L}$ extension 
of the standard models. 
In phenomenological results, the signal 
strengths for 
$h\rightarrow \ell \bar{\ell}\gamma$ at 
Large Hadron Collider and 
for $e^-e^+\rightarrow h\gamma$
at future Lepton Colliders
are analyzed in physical 
parameter space for both 
vector and chiral $B-L$ models. 
We find that the contributions 
from neutral gauge boson $Z'$ 
to the signal strengths are 
rather small. 
Consequence, the effects are 
hard to probe at future colliders.
While the impacts 
of charged Higgs in the 
chiral $B-L$ model on 
the signal strengths
are significant and 
can be measured with the help 
of the initial polarization 
beams at 
future lepton colliders.Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Science and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd.