  It has been suggested that Einstein's theory of General Relativity can be
modified to accomodate mismatches between the gravitational field and luminous
matter on a wide range of scales. Covariant theories of modified gravity
generically predict the existence of extra degrees of freedom which may be
interpreted as dark matter. We study a subclass of these theories where the
overall energy density in these extra degrees of freedom is subdominant
relative to the baryon density and show that they favour the presence of
massive neutrinos. For some specific cases (such as a flat Universes with a
cosmological constant) one finds a conservative lower bound on the neutrinos
mass of $m_\nu>0.31$ eV.
