Let $f :X \to Y$ be a submersion between smooth projective varieties over $\mathbb{C}$ and let $\alpha \in Z^k(X)$ be an algebraic cycle of $X$. Is is true that for all odd numbers $p$ and $q$ such that $p+q = 2k$, $\alpha$ lives in the kernel of $H^{2k}(X,\mathbb{Q}) \to H^p(Y,R^qf_*\mathbb{Q})$?

The reason why I ask this question is the following:

Since $f$ is a submersion, the class $[\alpha]$ in $H^p(Y,R^qf_*\mathbb{Q})$ can be represented by an element in

$$f^*\Omega^p_{Y,\mathbb{R}} \otimes \Omega^{q-p}_{X/Y,\mathbb{R}}.$$

However, it seems to me that any class which can be represented by one of above elements cannot be of type $(k,k)$. Can someone explain what happens here? Thank you.