Suppose $Y$ is a singular hypersurface in $\mathbb{P}^n$ whose singular locus of codimension $ \geq 2$, then the canonical sheaf $\omega_Y$ of $Y$ could still be defined, see
https://mathoverflow.net/questions/133253/how-to-define-the-canonical-sheaf-on-singular-varieties
Does adjunction formula still work now? i.e.
\begin{equation} \omega_Y \stackrel{?}{=} \omega_{\mathbb{P}^n} \otimes \mathcal{O}(Y)|_Y \end{equation}
More generally does adjunction formula work for general pairs $(Y,X)$ where $X$ is a smooth variety over a field $k$ and $Y$ is a singular hypersurface with singular locus of codimension $ \geq 2$?