An effective-field theory with correlations is developed for a mixed spin-1 and spin-5/2 Ising ferrimagnetic system on the honeycomb (delta = 3) and square (delta = 4) lattices in the absence and presence of a longitudinal magnetic field. The ground-state phase diagram of the model is obtained in the longitudinal magnetic field (h) and a single-ion potential or crystal-field interaction (Delta) plane. We also investigate the thermal variations of the sublattice magnetizations, and present the phase diagrams in the (Delta/|J|, k(B)T/|J|) plane. The susceptibility, internal energy and specific heat of the system are numerically examined, and some interesting phenomena in these quantities are found due to the absence and presence of the applied longitudinal magnetic field. Moreover, the system undergoes second- and first-order phase transition; hence, the system gives a tricritical point. The system also exhibits reentrant behavior.