  We consider a gravitational theory with an additional non-minimal coupling
between baryonic matter fields and geometry. The coupling is second order in
the energy momentum tensor and can be seen as a generalization of the
energy-momentum squared gravity model. We will add a constraint through a
Lagrange multiplier to ensure the conservation of the energy-momentum tensor.
Background cosmological implications together with its dynamical system
analysis will be investigated in details. Also we will consider the growth of
matter perturbation at first order, and estimate the model parameter from
observations on $H$ and also $f\sigma_8$. We will show that the model parameter
should be small and positive in 2$\sigma$ confidence interval. The theory is
shown to be in a good agreement with observational data.
