I think what you are looking for is the Lefschetz hyperplane theorem for singular varieties from Goresky and MacPherson's " Stratified Morse theory" (part II, section 1.2). The range in which there is an isomorphism depends on the number of equations needed to define $X$ locally. The theorem says (after some deciphering) that if this number is $\leq k$ for points of $X$ outside the hyperplane, then the hyperplane section map is an isomorphism in degrees up to $N-k-1$ where $N$ is the dimension of the ambient projective space.

Also, for middle perversity intersection homology the Lefschetz theorem is stated almost exactly the way as for smooth varieties and ordinary homology: for a generic hyperplane the hyperplane section map is an isomorphism in degrees $<\dim X-1$ and is surjective in degree $\dim X-1$.