Cycles contained in ample enough hypersurfaces

Question

Let $X$ be an irreducible smooth projective variety of pure dimension $d$ over the complex numbers and $Z\subset X\times X$ a codimension $d$ irreducible smooth closed subvariety.

Is there a smooth hyperplane section $H$ on $X$ such that $H\times X$ contains a cycle in the rational equivalence class of $Z$?

I expect the answer to be "no". Is it "yes" if one drops smoothness of $H$?

Answer 1

Let $X:=\Sigma_g$ be a smooth curve of genus $g\geq 2$ and $Z=\Delta$, where $\Delta \subset \Sigma_g \times \Sigma_g$ is the diagonal.

Then $\Delta$ is not rationally equivalent to a union of fibres of $p_1 \colon \Sigma_g \times \Sigma_g \to \Sigma_g$, for instance because $\Delta^2=2-2g <0$.