Roughly, I'd like to ask how does the first terms in de Rham complex behaves for singular varieties.

Let $Y$ be a potentially singular integral scheme over a perfect field $k$ of characteristic $p$ and $F : Y \to Y$ be its Frobenius morphism. The first mapping in de Rham complex of $Y$ (after application of $F_*$) is just the differential:
$$d : F_*\mathcal{O}_Y \to F_*\Omega^1_{Y/k}.$$
Its kernel clearly contains the image of the Frobenius morphism $F : \mathcal{O}_Y \to F_*\mathcal{O}_Y$. What are the sufficient assumptions for which the kernel $\textrm{Ker} (d)$ is in fact given by $F$?

For now, I can prove it for normal $Y$ with a torsion-free differential module $\Omega^1_Y$. The proof, more or less, boils down to localization at the generic point (which is injective by torsion-freeness) and then using normality ($a^p = f$ for $a \in k(Y)$ and $f \in \mathcal{O}_Y$ implies $a \in \mathcal{O}_Y$).

Is there any well-known, explicit class of varieties for which the kernel is bigger?

Maybe, there is a large class of normal singularities for which $\Omega^1_Y$ is torsion-free?
Do you know any reference for this kind of problems?