  We study Maurer-Cartan elements on homotopy Poisson manifolds of degree $n$.
They unify many twisted or homotopy structures in Poisson geometry and
mathematical physics, such as twisted Poisson manifolds, quasi-Poisson
$\g$-manifolds, and twisted Courant algebroids. Using the fact that the dual of
an $n$-term $L_\infty$-algebra is a homotopy Poisson manifold of degree $n-1$,
we obtain a Courant algebroid from a $2$-term $L_\infty$-algebra $\g$ via the
degree $2$ symplectic NQ-manifold $T^*[2]\g^*[1]$. By integrating the Lie
quasi-bialgebroid associated to the Courant algebroid, we obtain a
Lie-quasi-Poisson groupoid from a $2$-term $L_\infty$-algebra, which is
proposed to be the geometric structure on the dual of a Lie $2$-algebra. These
results lead to a construction of a new 2-term $L_\infty$-algebra from a given
one, which could produce many interesting examples.
