I am looking for an example (if such exist) of a smooth projective variety $X$ whose $\mathbb{Q}$-homology is generated by algebraic cycles, and yet does not have a second homotopy group, $\pi_2(X)=0.$ Thus, algebraic cycles that span $H_2(X,\mathbb{Q})$ are coming from some non-rational curves.

I am looking for an example (if such exist) of a smooth projective variety $X$ whose $\mathbb{Q}$-homology $H_*(X,\mathbb{Q})$ is generated by algebraic cycles, and yet does not have a second homotopy group, $\pi_2(X)=0.$ Thus, algebraic cycles that span $H_2(X,\mathbb{Q})$ are coming from some non-rational curves.