Suppose that $M$ is a symplectic manifold with a Hamiltonian circle action. Is there a topological Lefschetz pencil on $M$, $f\colon M-A \rightarrow S^2$, such that the fibers are symplectic submanifolds of $M$ and such that the circle action restricts to a Hamiltonian circle action on the fibers?

(After reading Tim's answer.)

Are there some cases when there do exist such things?