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} $.