Effective density of values of indefinite ternary inhomogeneous quadratic forms

Given an inhomogeneous quadratic form $Q_\xi(v)=Q(v+\xi)$ with $Q$ an indefinite $\mathbb{Q}$-isotropic rational ternary form and $\xi\in \mathbb{R}^3$ irrational, we prove an effective lower bound for the number of integer vectors $v\in \mathbb{Z}^n$ with $\|v\| \leq T$ such that $|Q_\xi(v)-t|<\delta$ that is valid for any $t\in \mathbb{R}$ and all $\delta\geq T^{-\nu}$, with $\nu>0$ depending explicitly on the Diophantine properties of $\xi$. In particular, for $\xi$ with algebraic entries we can take any $\nu<\frac{1}{8}$.