One of the most critical issues in relativistic astrophysics is explaining the origin mechanisms of (ultra)high-energy charged particle components of cosmic rays. Black holes (BHs), which are vast reservoirs of (gravitational) energy, are candidates for such energetic cosmic ray sources. The main idea of this study is to investigate the effects of scalar-tensor-vector gravity (STVG) and so-called modified gravity (MOG) on charged particle acceleration by examining their dynamics and acceleration through the magnetic Penrose process (MPP) near magnetized Kerr BHs in MOG (Kerr-MOG BHs). First, we briefly study the horizon structure of the Kerr-MOG BH. Then, we derive the effective potential for the circular motion of charged particles by considering electromagnetic and MOG field interactions on the particles to gain insight into the stability of circular orbits. Our results show that the magnetic field can extend the region of stable circular orbits, whereas the STVG parameter reduces the instability of the circular orbit. Thus, from the examination of particle trajectories, we observe that, at fixed values of other parameters, the Schwarzschild BH captures the test particle; in the case of the Kerr BH, the test particle escapes to infinity or is captured by the BH, while in the Kerr-MOG BH, the test particle is trapped in some region around the BH and starts orbiting it at a smaller value of the MOG field parameter. By investigating the MPP, we found that, in stronger magnetic fields, the behavior of orbits becomes more chaotic. As a result, the particle escapes to infinity with high energies.