The so-called Harada gravity with non-conserved energy momentum tensor is here taken into account. It includes Rastall gravity as a special case. The field equations are written as Einstein equations where the source is supplemented by a divergence-free conformal Killing tensor and a tensor proportional to the metric, linear in the scalar curvature and the trace of the energy-momentum tensor. These terms can be natural candidates for dark sector and give rise to a coupling of the dark sector with the matter content. The field equations are the conformal Killing extension of Rastall gravity, and include Unimodular gravity. In a Friedmann-Robertson-Walker (FRW) background, the Cosmic Microwave Background (CMB) restricts parameters so that the dark sector only couples with the trace of the stress-energy tensor. The explicit form of the tensor for the dark sector is found, and the Friedmann and continuity equations are presented, with a standard cosmological analysis. The sum of energy momentum tensors of dust matter and of dark fluid is conserved, and the dust energy density evolves as ${\mu }_M\propto a^{-3/(1+\tau )}$, with an exponent modified by the coupling $$ with the dark fluid.