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An electron-positron cascade in the magnetospheres of Kerr black holes (BHs) is a fundamental ingredient to fueling the relativisticγ-ray jets seen at the polar regions of galactic supermassive BHs (SMBHs). This leptonic cascade occurs in thespark gapregion of a BH magnetosphere where the unscreened electric field parallel to the magnetic field is present; hence, it is affected by the magnetic field structure. A previous study explored the case of a thin accretion disk, representative of active galactic nuclei. Here we explore the case of a quasi-spherical gas distribution, as is expected to be present around the SMBH Sgr A* in the center of our Milky Way galaxy, for example. The properties and efficiency of the leptonic cascade are studied. The findings of our study and the implications for SMBH systems in various spectral and accretion states are discussed. The relationships and scalings derived from varying the mass of the BH and background photon spectra are further used to analyze the leptonic cascade process to power jets seen in astronomical observations. In particular, one finds the efficiency of the cascade in a quasi-spherical gas distribution peaks at the jet axis. Observationally, this should lead to a more prominent jet core, in contrast to the thin disk accretion case, where it peaks around the jet–disk interface. One also finds the spectrum of the background photons plays a key role. The cascade efficiency is maximum for a spectral index of 2, while harder and softer spectra lead to a less efficient cascade.