  We show the existence of a hypersurface that contains a given closed
subscheme of a projective space over a finite field and intersects a smooth
quasi-projective scheme smoothly, under some condition on the dimension. This
generalizes a Bertini theorem by Poonen and is the finite field analogue of a
Bertini theorem by Altman and Kleiman. Furthermore, we add the possibility of
modifying finitely many local conditions of the hypersurface. We show that the
condition on the dimension is fulfilled for schemes with simple normal
crossings and give an application to embeddings into smooth schemes.
