for completeness let me add that, in the case of twisted Poisson structures, if $i_{\mathcal{O}}:\mathcal{O}\to M$ is one of the induced leaves of the foliation, then $$i_{\mathcal{O}}^*(\phi)=d\Omega_{\mathcal{O}}$$ where $\Omega_{\mathcal{O}}$ is the 2-form inverse to the restriction of $\Pi$ to the foliation. In case $\phi=0$ we are in the Poisson case and $\Omega_{\mathcal{O}}$ is the symplectic form on the symplectic leaf $\mathcal{O}$.