  We characterize normal $3$-pseudomanifolds with $g_2\leq4$. We know that if a
$3$-pseudomanifold with $g_2\leq4$ does not have any singular vertices then it
is a $3$-sphere. We first prove that a normal $3$-pseudomanifold with
$g_2\leq4$ has at most two singular vertices. Then we prove that a normal
$3$-pseudomanifold with $g_2 \leq 4$, which is not a $3$-sphere is obtained
from some boundary of $4$-simplices by a sequence of operations connected sum,
edge expansion and an edge folding. In addition, by using [17], we re-framed
the characterization of normal $3$-pseudomanifolds with $g_2\leq 9$, when it
has no singular vertices.
