Let $\mathrm{\Gamma }$ be a group, $\mathcal{A}$ be a $\mathrm{\Gamma }$-graded commutative ring with unity $1,$ and $\mathcal{D}$ a graded $\mathcal{A}$-module. In this paper, we introduce the concept of graded weakly $J_{gr}$-semiprime submodules as a generalization of graded weakly semiprime submodules. We study several results concerning of graded weakly $J_{gr}$ -semiprime submodules. For example, we give a characterization of graded weakly $J_{gr}$-semiprime submodules. Also, we find some relations between graded weakly $J_{gr}$-semiprime submodules and graded weakly semiprime submodules. In addition, the necessary and sufficient condition for graded submodules to be graded weakly $J_{gr}$-semiprime submodules are investigated. A proper graded submodule $U$ of $\mathcal{D}$ is said to be a graded weakly $J_{gr}$-semiprime submodule of $\mathcal{D}$ if whenever $r_g\in h(\mathcal{A}),$ $m_h\in h(\mathcal{D})$ and $n\in {\mathbb{Z}}^{+}$ with $0\ne r_g^n{m}_h\in U$, then $r_g{m}_h\in U+J_{gr}(\mathcal{D})$, where $J_{gr}(\mathcal{D})$ is the graded Jacobson radical of $\mathcal{D}.$

Let $R$ be a $G$ graded commutative ring and $M$ be a $G$-graded $R$-module. The set of all graded second submodules of $M$ is denoted by $Spe{c}_G^s(M),$ and it is called the graded second spectrum of $M$. We discuss graded rings with Noetherian graded prime spectrum. In addition, we introduce the notion of the graded Zariski socle of graded submodules and explore their properties. We also investigate $Spe{c}_G^s(M)$ with the Zariski topology from the viewpoint of being a Noetherian space.