  Let $\psi:\mathbb R_{+}\to \mathbb R_{+}$ be a nonincreasing function. A pair
$(A,\mathbf b),$ where $A$ is a real $m\times n$ matrix and $\mathbf
b\in\mathbb R^{m},$ is said to be $\psi$-Dirichlet improvable, if the system
$$\|A\mathbf q +\mathbf b-\mathbf p\|^m<\psi(T), \quad \|\mathbf q\|^n<T$$ is
solvable in $\mathbf p\in\mathbb Z^{m},$ $\mathbf q\in\mathbb Z^{n}$ for all
sufficiently large $T$ where $\|\cdot\|$ denotes the supremum norm. For
$\psi$-Dirichlet non-improvable sets, Kleinbock--Wadleigh (2019) proved the
Lebesgue measure criterion whereas Kim--Kim (2021) established the Hausdorff
measure results. In this paper we obtain the generalised Hausdorff $f$-measure
version of Kim--Kim (2021) results for $\psi$-Dirichlet non-improvable sets.
