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 $$\mathfrak g$$ g -manifolds, and twisted Courant algebroids. Using the fact that the dual of an n-term $$L_\infty $$ L ∞ -algebra is a homotopy Poisson manifold of degree $$n-1$$ n - 1 , we obtain a Courant algebroid from a 2-term $$L_\infty $$ L ∞ -algebra $$\mathfrak g$$ g via the degree 2 symplectic NQ-manifold $$T^*[2]\mathfrak g^*[1]$$ 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 $$L_\infty $$ L ∞ -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 $$L_\infty $$ L ∞ -algebra from a given one, which could produce many interesting examples.