Conducted by the recent progress in new carbonaceous materials with innovative properties and the remarkable applicability as allotropic graphenes, particularly the biphenylene network (BPN), constituted by the combination of four, six, and eight rings, we treat this structure using a multi-spin Ising model through a Monte Carlo simulation (MCS). We meticulously explore the compensation and critical behaviors of a mixed spin σ = ± 3/2, ± 1/2 and spin S = ± 1, 0. By analyzing the impact of the exchange couplings (JσS, JS), particularly the cyclic coupling, denoted as Jcyc, along with the crystal fields (DS, Dσ) and external magnetic field h, we unveil appealing ground-state (T = 0) phase diagrams, in addition to the magnetic properties covering magnetizations, magnetic susceptibility, and internal energy. The insertion of Jcyc is advantageous for the emergence of Tcomp, with possibly two compensation points that manifest for negative values of Jcyc exceeding a designated threshold value of JσS. The crystal fields constantly culminate in second-order phase transitions, wherein the compensation behavior exists exclusively when Dσ lowers negatively. Introducing Jcyc enables us to obtain a maximum number of magnetic plateaus within the hysteresis loop when varying the magnetic field h. The variation of the Hamiltonian parameters entails only a single-loop phenomenon; however, the crystal field Dσ may result in triple loops.