  The accelerated expansion of the universe is a rather established fact in
cosmology and many different models have been proposed as a viable explanation.
Many of these models are based on the standard general relativistic framework
of non-interacting fluids or more recently of coupled (interacting) dark energy
models, where dark energy (the scalar field) is coupled to the dark matter
component giving rise to a fifth-force. An interesting alternative is to couple
the scalar field directly to the gravity sector via the Ricci scalar. These
models are dubbed non-minimally coupled models and give rise to a
time-dependent gravitational constant. In this work we study few models falling
into this category and describe how observables depend on the strength of the
coupling. We extend recent work on the subject by taking into account also the
effects of the perturbations of the scalar field and showing their relative
importance on the evolution of the mass function. By working in the framework
of the spherical collapse model, we show that perturbations of the scalar field
have a limited impact on the growth factor (for small coupling constant) and on
the mass function with respect to the case where perturbations are neglected.
