  Characterizing face-number-related invariants of a given class of simplicial
complexes has been a central topic in combinatorial topology. In this regard,
one of the well-known invariants is $g_2$. Let $K$ be a normal
$3$-pseudomanifold such that $g_2(K) \leq g_2(lk (v)) + 9$ for some vertex $v$
in $K$. Suppose either $K$ has only one singularity or $K$ has two
singularities (at least) one of which is an $\mathbb{RP}^2$-singularity. We
prove that $K$ is obtained from some boundary complexes of $4$-simplices by a
sequence of operations of types connected sums, bistellar $1$-moves, edge
contractions, edge expansions, vertex foldings, and edge foldings. In case $K$
has one singularity, $|K|$ is a handlebody with its boundary coned off.
Further, we prove that the above upper bound is sharp for such normal
$3$-pseudomanifolds.
