Lefschetz (1,1) theorem for quasi-projective varieties

Question

I was reading about Hodge conjecture on Wikipedia but it started with the assumption that $X$ is smooth projective.

If $X$ is a smooth quasi-projective variety, then corresponding to smooth sub-varieties of $X$ of codimension 1, one gets a class in $H^2(X,\mathbb{Q})$. This cohomology also carries a Mixed Hodge structure. One should be able to phrase Hodge conjecture for such varieties.

My question is

1. Does it make sense to talk about Hodge conjecture for smooth(normal?) quasi-projective varieties? If yes, what would be the desired statement?

2. Can one prove it in the easiest of the cases? For example : lefschetz (1,1) theorem is the base case. If yes, proofs/references are most welcome.

Thanks!