  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.
