We investigate some critical phenomena of a Ising ferrimagnetic system on a square lattice with spins $Q=\pm 7/2, \pm 5/2, \pm 3/2$ , and ±1/2 in a $B$ sublattice and spins $S=\pm 5/2, \pm 3/2$ , and ±1/2 in a $A$ sublattice in presence of an $h'$ external magnetic field, such that $Q$ and $S$ are the nearest-neighbors ferrimagnetically coupled. Using Monte Carlo simulations, we calculate the finite-temperature phase diagrams of magnetization, specific heat, and magnetic susceptibility of the model. The system is defined by a Hamiltonian $(\mathcal {H})$ that contains ferromagnetic next-nearest-neighbors interactions between $S$ spins ( $J'_{2}$ ), as well as external magnetic field and anisotropy interactions at each point of lattice. For a defined range of parameters in $\mathcal {H}$ , the system exhibits discontinuities in their physical variables and spin compensation behaviors. In the planes $J'_{2}-k_{\mathrm{ B}}T'$ and $h'-k_{\mathrm{ B}}T'$ , we analyze the relationship between the critical, compensation, and discontinuous phase transition temperatures with the external magnetic field and the next-nearest-neighbors interaction between $S$ spins. We found that the existence of discontinuous phase transition depends on the strengths of $h'$ and $J'_{2}$ .