Bertini type result for complete intersection varieties containg a non-singular variety

Let $$X$$ be a smooth, projective $$\mathbb{C}$$-variety of dimension $$r$$ containing in $$\mathbb{P}^n$$. Denote by $$I_X \subset \mathbb{C}[X_0,...,X_n]$$ the ideal of $$X$$ defined by some homogeneous polynomials. For $$k \gg 0$$ and a general choice of $$n-r$$ homogeneous polynomials $$F_1,...,F_{n-r} \in I_X$$ of degree $$k$$, I want to understand the singularities of the complete intersection variety defined by $$F_1,...,F_{n-r}$$? In particular, is it possible that the complete intersection variety is a union of smooth projective varieties, intersecting each other transversally?

I would imagine this is too much to expect, so is there some criterion on $$X$$, when the complete intersection variety has such nice singularities? I would very much like to know a reference which studies similar questions.

The general such intersection is the union $$X \cup Y$$, where $$Y$$ is smooth away from $$X$$. On the other hand, in general $$Y$$ has singularities in codimension 4. Indeed, if $$\mathcal{I}_X$$ is the ideal sheaf of $$X$$, the polynomials $$F_i$$ induce a morphism $$\mathcal{O}^{\oplus (n-r)} \to \mathcal{I}_X(k).$$ When restricted to $$X$$ it is a morphism $$\mathcal{O}_X^{\oplus (n-r)} \to \mathcal{I}_X/\mathcal{I}_X^2(k) \cong \mathcal{N}_X^\vee(k)$$ of two locally free sheaves of rank $$k$$. Its degeneracy locus is the intersection $$X \cap Y$$, and its second degeneracy locus (the corank-2 locus) is the singular locus of $$Y$$.