  In this paper, we investigate the gravitational effects of a global monopole
that couples nonminimally to gravity. Considering a coupling parameter of
arbitrary strength, we have obtained an analytical solution for the field
equations of the model in the asymptotic region outside the monopole's core,
thus generalizing previous results. Within the weak coupling regime, we have
also analyzed how this matter-geometry interplay affects some physical
properties of the monopole, such as its radius and the mass enclosed in its
core, which are explicitly computed and expressed in terms of the parameter of
the theory. Using the Hahari-Loust\'o toy model, we have also found that the
modification of general relativity (GR) may render a positive sign to the
monopole mass, which means a global monopole able to exert attractive
gravitational force on matter particles surrounding it. Next, we have studied
some properties of a hairy black hole carrying such a global monopole charge.
In this vein, we verified that the deviation from GR may provide such a
spacetime with both event and Cauchy horizons, thus emulating a
Reissner-Nordstr\"om framework. Finally, we have performed a study of the
geodesic motion of timelike particles around this black hole by setting the
conditions for stable circular orbits. We also examined the gravitational
bending experienced by photons traveling through this geometry, taking into
account finite-distance corrections on the deflection angle. Additionally, we
make an order-of-magnitude estimate for this extra contribution considering the
light bending caused by Sagittarius A*.
