An algebraic correspondence between two varieties, $V,W$ is a kind of multi-valued map from $V$ to $W$, or, in other words, a map from a covering $U$ of $V$ to $W$. Apparently, such a map gives a morphism from the cohomology of $W$ to that of $V$. How does this work? I know that we would get a map to the cohomology of $U$, by the functorial nature of cohomology. However, how do we then map this to an element of the cohomology of $V$?
I'm going to define correspondence a little differently. I usually think of a correspondence as a closed subset $Z\subset V\times W$ maybe with some extra properties. This should be the same as what you wrote with the property that the projection on the first factor is actually a covering.
The way I get a map on cohomology is to take the class of this subvariety $[Z]\in H^*(V\times W)$ (pick any cohomology theory you want with a cycle class map). The degree it sits in will depend on codimension.
Let $p: V\times W\to V$ and $q:V\times W\to W$ be the projections. Now we can use pullback and pushforward and the ring structure on the cohomology to get a map $\Phi: H^*(V)\to H^*(W)$ by first pulling back $p^* :H^*(V)\to H^*(V\times W)$, then cupping with the class $\cup [Z] : H^*(V\times W)\to H^*(V\times W)$ then pushing forward $q_*: H^*(V\times W)\to H^*(W)$.
Thus the map on cohomology is $\Phi (\alpha)= q_*(p^*\alpha \cup [Z])$.