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Injective covers of modules were introduced by Enochs [5] as a categorical dual of an injective envelope and they have been investigated by many authors. The T-torsionfree injective covers have been studied in [I]. The relative cases, for injective modules relative to a torsion theory, were introduced in [18]. In this paper C-covers are studied in the general setting of a Grothendieck category with the aim of applying it to relative injective covers in the category of left R-modules and the category of graded R-modules, where R is a G-graded ring. The Section 2 is of preparatory nature, we introduce the concepts and auxiliary results. In Section 3, we characterize C-covers in terms of C-pure sequences (Proposition 5); when C is the class of T-torsionfree T-injective R-modules we obtain a more precise result. In Section 4 we induce precovers by using adjoint functors, and study the graded case and polynomial rings …