Partially magnetized low-temperature plasmas (LTP) in an E×B configuration, where the applied magnetic field is perpendicular to the self-consistent electric field, have become increasingly relevant in industrial applications. Hall thrusters, a type of electrostatic plasma propulsion, are one of the main LTP technologies whose advancement is hindered by the not-fully-understood underlying physics of operation, particularly, with respect to the plasma instabilities and the associated electron cross-field transport. The development of Hall thrusters with unconventional magnetic field topologies has imposed further questions regarding the instabilities’ characteristics and the electrons’ dynamics in these modern cross-field configurations. Accordingly, we present in this effort a detailed parametric study of the influence of three factors on the plasma processes in the radial-azimuthal coordinates of a Hall thruster, namely, the magnetic field gradient, Secondary Electron Emission, and plasma number density. The studies are carried out using the reduced-order particle-in-cell (PIC) code developed by the authors. The setup of the radial-azimuthal simulations largely follows a well-defined benchmark case from the literature in which the magnetic field is oriented along the radius and a constant axial electric field is applied perpendicular to the simulation plane. The salient finding from our investigations is that, in the studied cases corresponding to elevated plasma densities, a long-wavelength azimuthal mode with the frequency of about 1 MHz is developed. Moreover, in the presence of strong magnetic field gradients, this mode results from an inverse energy cascade and induces a significant electron cross-field transport as well as a notable heating of the ions.