  The effective photon-quark-antiquark ($\gamma q \overline{q}$) vertex
function is evaluated at finite temperature in presence of an arbitrary
external magnetic field using the 2-flavour gauged Nambu--Jona-Lasinio (NJL)
model in the mean field approximation. The lowest order diagram contributing to
the magnetic form factor and the anomalous magnetic moment (AMM) of the quarks
is calculated at finite temperature and external magnetic field using the
imaginary time formalism of finite temperature field theory and the Schwinger
proper time formalism. The Schwinger propagator including all the Landau levels
with non-zero AMM of the dressed quarks is considered while calculating the
loop diagram. Using sharp as well as smooth three momentum cutoff, we
regularize the UV divergences arising from the vertex function and the
parameters of our model are chosen to reproduce the well known phenomenological
quantities at zero temperature and zero magnetic field, such as pion-decay
constant ($f_\pi$), vacuum quark condensate, vacuum pion mass ($m_\pi$) as well
as the magnetic moments of proton and neutron. We then study the temperature
and magnetic field dependence of the AMM and constituent mass of the quark. We
found that, the AMM as well as the constituent quark mass are large at the
chiral symmetry broken phase in the low temperature region. Around the
pseudo-chiral phase transition they decrease rapidly and at high temperatures
both of them approach vanishingly small values in the symmetry restored phase.
