For a projective variety $X$ in ${\mathbb{P}}^m$ of dimension $n$, an additive action on $X$ is an effective action of ${\mathbb{G}}_a^n$ on ${\mathbb{P}}^m$ such that $X$ is ${\mathbb{G}}_a^n$-invariant and the induced action on $X$ has an open orbit. Arzhantsev and Popovskiy have classified additive actions on hyperquadrics of corank 0 or 1. In this paper, we give the classification of additive actions on hyperquadrics of corank 2 whose singularities are not fixed by the ${\mathbb{G}}_a^n$-action.