Let $Δ$ be a $g_2$-minimal normal 3-pseudomanifold. A vertex in $Δ$ whose link is not a sphere is called a singular vertex. When $Δ$ contains at most two singular vertices, its combinatorial characterization is known [9]. In this article, we present a combinatorial characterization of such a $Δ$ when it has three singular vertices, including one $\mathbb{RP}^2$-singularity, or four singular vertices, including two $\mathbb{RP}^2$-singularities. In both cases, we prove that $Δ$ is obtained from a one-vertex suspension of a surface, and some boundary complexes of $4$-simplices by applying the combinatorial operations of types connected sums, vertex foldings, and edge foldings.