The asserted set $x$ is just a set that contains all overlap sets between the set $a$ and any set, among its elements. Weak Power as written above is simply:
$(\forall a)(\exists x)(\forall y)(\exists z\in x) (z= a \cap y)$
Now in classical ZF all subsets of $a$ are overlaps with $a$, so all of them would be included in the weak power of $a$. Then by separation one can easily recover $P(a)$ axiom by separating on the weak power of $a$ using the property of being a subset of $a$.