Example II.$3.2.6$ in Hartshorne (reduced induced closed subscheme structure)
This question is essentially the same as mine but it seems to have a rather complicated answer without upvotes.
Basically in this example Hartshorne says that you can reduce the problem of proving that the glueing properties hold for the reduced, induced closed subscheme of $Y$ for an affine cover $\{U_i\}$ of $X$ to showing that given affine open $U= \operatorname{Spec}(A)$ and $f \in A$, show that the reduced structure on $D(f) \cap Y$ induced by the restriction of the reduced structure on $\operatorname{Spec}(A) \cap Y$ is the same as the reduced structure on $\operatorname{Spec}(A_f) \cap Y$.
I can see it for the case that $Y \cap U_i \cap U_j$ is both $Y \cap U_i \cap D(f)$ w.r.t $U_i$ and $Y \cap D(g) \cap U_j$ w.r.t $U_j$ as then we can apply this result directly but if not it seems to me that we need a further glueing result to say that, since $U_i \cap U_j$ is open in both $U_i$ and $U_j$ then it is the union of some $D(f)$ in both, and then try to apply this result to the union.
If this is unclear which I feel it might be I will try and add some more details.
