Let $R$ be a commutative Noetherian ring and let $C$ be a semidualizing $R$-module. We prove a result about the covering properties of the class of relative Gorenstein injective modules with respect to $C$ which is a generalization of Theorem 1 by Enochs and Iacob (2015). Specifically, we prove that if for every $G_C$-injective module $G$, the character module $G^{+}$ is $G_C$-flat, then the class ${\mathrm{𝒢ℐ}}_C\left(R\right)\cap {𝒜}_C\left(R\right)$ is closed under direct sums and direct limits. Also, it is proved that under the above hypotheses the class ${\mathrm{𝒢ℐ}}_C\left(R\right)\cap {𝒜}_C\left(R\right)$ is covering....