Let $X\subset\mathbb{P}^n$ be e normal variety, $L\subset\mathbb{P}^n$ a linear subspace, and $Y = X\cap L$ a linear section. Assume that $Y$ is also normal. In particular, we have that $Sing(X)$ has codimension at least two in $X$, and $Sing(Y)$ has codimension at least two in $Y$.

Does there exist any generalization of the Lefschetz hyperplane theorem to higher codimension linear sections of singular varieties ensuring that the restriction map

$$Cl(X)\rightarrow Cl(Y)$$

is an isomorphism or at least surjective?