The singular locus of hypersurface sections containing a closed subscheme over finite fields

We prove that there exist hypersurfaces that contain a given closed subscheme $Z$ of the projective space over a finite field and intersect a given smooth scheme $X$ off of $Z$ smoothly, if the intersection $V = Z \cap X$ is smooth. Furthermore, we can give a bound on the dimension of the singular locus of the hypersurface section and prescribe finitely many local conditions on the hypersurface. This is an analogue of a Bertini theorem of Bloch over finite fields and is proved using Poonen's closed point sieve. We also show a similar theorem for the case where $V$ is not smooth.