  Various thermodynamic quantities and the phase diagram of strongly
interacting hot and dense magnetized quark matter are obtained with the $ 2
$-flavour Nambu-Jona-Lasinio model with Polyakov loop considering finite values
of the anomalous magnetic moment (AMM) of the quarks. Susceptibilities
associated with constituent quark mass and traced Polyakov loop are used to
evaluate chiral and deconfinement transition temperatures. It is found that,
inclusion of the AMM of the quarks in presence of the background magnetic field
results in a substantial decrease in the chiral as well as deconfinement
transition temperatures in contrast to an enhancement in the chiral transition
temperature in its absence. Using standard techniques of finite temperature
field theory, the two point thermo-magnetic mesonic correlation functions in
the scalar ($\sigma$) and neutral pseudoscalar ($\pi^0$) channels are evaluated
to calculate the masses of $\sigma $ and $ \pi^0 $ considering the AMM of the
quarks.
