  Let $\Delta$ be a $g_2$-minimal normal 3-pseudomanifold. A vertex in $\Delta$
whose link is not a sphere is called a singular vertex. When $\Delta$ contains
at most two singular vertices, its combinatorial characterization is known [9].
In this article, we present a combinatorial characterization of such a $\Delta$
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 $\Delta$ 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.
