Given a complex quasi-projective variety $X$, let $\alpha$ be an element of the Borel–Moore homology $H_i^\text{BM}(X)$ such that it can be killed by a prime $p$. Under what conditions one can say that there is a closed subvariety of $Z$ such that $\alpha$ is the pushforward of an element in the homology of $Z$? and under what conditions one can say the homology class on $Z$ is torsion?

This question is inspired by the fact that with the assumption of smoothness, if $\alpha$ is a $p$-torsion element in the cohomology of $X$ then there is a divisor $D$ such that pullback of $\alpha$ to $X-D$ is zero. This follows from Bloch–Kato.