We construct a Dvali–Gabadadze–Porrati (DGP)-inspired braneworld scenario where a scalar field non-minimally coupled to the induced Ricci curvature is present on the brane. First we show that this model allows for an embedding of the standard Friedmann cosmology in the sense that the cosmological evolution of the background metric on the brane can be described by the standard Friedmann equation plus energy non-conservation on the brane. Then cosmological implications of this scenario are examined in detail and the quintessence model and late-time expansion of the universe within this framework are discussed. Some observational constraints imposed on this non-minimal scenario are studied and the relation of this model to the dark radiation formalism is determined.