Interplay of dark matter and plasma in shaping black-hole shadows and lensing in Horndeski gravity

We investigate black-hole shadow and weak gravitational lensing observables for a static, spherically symmetric non-minimally coupled Horndeski black hole surrounded by perfect fluid dark matter (PFDM), taking into account the dispersive propagation of photons in a plasma medium. The background geometry is described by the quartic square-root Horndeski solution supplemented by a PFDM logarithmic correction, and the corresponding black-hole and horizonless regions in parameter space are identified. Photon motion is studied in a cold, non-magnetized plasma, including the effect of gravitational redshift on the local photon frequency. For the shadow analysis, we consider both homogeneous and inhomogeneous power-law plasma distributions and determine the photonsphere and shadow radii numerically. In the weak-deflection regime, we derive the plasma-corrected bending angle, construct the lens equation, and evaluate the Einstein angle and image magnification, including the case of a singular isothermal sphere plasma profile. We further confront the shadow radius with observational bounds from the Sgr A * black-hole shadow and use the corresponding 1σ and 2σ intervals to delineate the allowed regions of the model parameter space. Our results show that the Horndeski parameter, the PFDM parameter, and plasma dispersion leave distinct and, in general, competing imprints on the photon-sphere radius, shadow size, deflection angle, and magnification. In particular, increasing the Horndeski parameter enlarges the photon sphere and shadow radius in the parameter range considered, whereas the PFDM contribution and plasma effects tend to suppress the shadow size and modify the weak-lensing observables in a frequency-dependent manner. These results indicate that environmental effects from dark matter and plasma should be modeled consistently when using shadow and lensing measurements to constrain deviations from general relativity.