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We consider an extension of standard General Relativity in which theHilbert-Einstein action is replaced by an arbitrary function of the Ricciscalar, nonmetricity, torsion, and the trace of the matter energy-momentumtensor. By construction, the action involves a non-minimal coupling betweenmatter and geometry. The field equations of the model are obtained, and theylead to the nonconservation of the matter energy-momentum tensor. Athermodynamic interpretation of the nonconservation of the energy-momentumtensor is also developed in the framework of the thermodynamics of theirreversible processes in open systems. The Newtonian limit of the theory isconsidered, and the generalized Poisson equation is obtained in the lowvelocity and weak fields limits. The nonmetricity, the Weyl vector, and thematter couplings generate an effective gravitational coupling in the Poissonequation. We investigate the cosmological implications of the theory for twodifferent choices of the gravitational action, corresponding to an additive anda multiplicative algebraic structure of the function $f$, respectively. Weobtain the generalized Friedmann equations, and we compare the theoreticalpredictions with the observational data. \te{We find that the cosmologicalmodels can give a good descriptions of the observations up to a redshift of$z=2$, and, for some cases, up to a redshift of $z=3$.

Non-minimal geometry-matter couplings in Weyl-Cartan space-times: $f(R,T,Q,T_m)$ gravity