The effect of spacetime curvature on optical properties may provide an opportunity to suggest new tests for gravity theories. In this paper, we investigated gravitational weak lensing around a Bardeen black hole with the string clouds parameter. First, we examined the horizon structure in the presence of string clouds around the gravitational compact object defined by Bardeen spacetime. The effect of gravitational weak lensing in a plasma medium is also discussed. According to the findings, the influence of the string cloud parameter on the circular orbits of a light ray around the black hole is greater than that in the Schwarzschild case, while the influence of the charge is reversed. The deflection angle of light rays in weak lensing is also used to study how much the image is magnified.

The motion of photons and the frequency shift of photons emitted by particles orbiting the central black hole described by the deformed Reissner–Nordström spacetime metric is investigated. It has been shown that in spacetime, two stable photon spheres are formed due to the gravitational attraction of photons by the central gravitating compact object. It has been found that as spacetime parameters increase, charge and deformation photon spheres become smaller in size and the effect of the electric charge is stronger than the effect of the deformation parameter. The frequency shift of photons radiated by the particles at the equatorial plane becomes stronger for the smaller values of the deformation parameter and the electric charge of the deformed Reissner–Nordström black hole.

In this research paper, we have examined the impact of a gravitomagnetic charge on extracting energy from a rotating Kerr-Taub-NUT black hole and Kerr-MOG (MOdified Gravity known as Scalar-Tensor-Vector-Gravity (STVG)) black hole embedded in the magnetic field via the magnetic Penrose process (MPP). Our findings demonstrate that the gravitomagnetic monopole and MOG parameters both significantly influence the extraction of energy from a rotating black hole, exceeding the MPP for the Kerr black hole in terms of energy efficiency. After comparing the results of this study with the observational data, we were confident that our findings on the amplification of MPP by MOG and NUT parameters are nearly in line with the observations of ultrarelativistic particles in the cosmic rays.

The paper has explored analogue of gravitational synchrotron massive particle and Penrose process in MOdified Gravity (MOG) known as Scalar-Tensor-Vector-Gravity (STVG). Investigation of the gravitational field around Kerr-MOG black hole showed that it has strong gravitational field with large horizon and can rotate faster than Kerr black hole due to the effect of STVG. We have studied influence of STVG in circular motion of massive particle around Kerr-MOG black hole and discussed the Innermost Stable Circular Orbit (ISCO) of massive test particle. It is shown that STVG plays a crucial role in energy extraction from a rotating black hole, with an energy efficiency of more than $100\%$ according to the Penrose process. Furthermore, we have explored the gravitational synchrotron radiation analogue produced by a massive particle orbiting around a Kerr-MOG black hole. It has been shown that the intensity of gravitational radiation from binary systems of stellar black holes (SBH) and supermassive black holes (SMBH).