  Employing a field dependent three-momentum cut-off regularization technique,
we study the phase structure and mesonic masses using the $2$-flavour
Nambu-Jona Lasinio model at finite temperature and density in presence of
arbitrary external magnetic field. This approach is then applied to incorporate
the effects of the anomalous magnetic moment(AMM) of quarks on constituent
quark mass and thermodynamic observables as a function of temperature/baryonic
density. The critical temperature for transition from chiral symmetry broken to
the restored phase is observed to decrease with the external magnetic field,
which can be classified as inverse magnetic catalysis, while an opposite
behaviour is realized in the case of a vanishing magnetic moment, implying
magnetic catalysis. These essential features are also reflected in the phase
diagram. Furthermore, the properties of the low lying scalar and neutral
pseudoscalar mesons are also studied in presence of a hot and dense magnetized
medium including AMM of the quarks using random phase approximation. For
non-zero values of magnetic field, we notice a sudden jump in the mass of the
Goldstone mode at and above the Mott transition temperature which is found to
decrease substantially with the increase in magnetic field when the AMM of the
quarks are taken into consideration.
