Given a symplectic manifold $(X,\omega)$, there is the group of Hamiltonian symplectomorphisms $HamSympl(X,\omega)$ acting on $X$. If $(X,\omega)$ is prequantizable this lifts to the group of quantomorphisms, both of them covering the diffeomorphisms of $X$:
quantomorphisms $\to$ Hamiltonian symplectomorphisms $\to$ diffeomorphisms .
A Hamiltonian action of a Lie group $G$ on $(X,\omega)$ is an action by quantomorphisms, hence a Lie group homomorphism $\hat \phi : G \to Quant(X, \omega)$
See (Brylinski, prop. 2.4.10).
In the literature this is usually discussed at the infinitesimal level, hence for the corresponding Lie algebras:
smooth functions+Poisson bracket $\to$ Hamiltonian vector fields $\to$ vector fields
Now an (infinitesimal) Hamiltonian action is a Lie algebra homomorphism $\mu : \mathfrak{g} \to (C^\infty(X), \{-,-\})$
Dualizing, the homomorphism $\mu$ is equivalently a linear map
which is a homomorphism of Poisson manifolds. This is called the moment map of the (infinitesimal) Hamiltonian $G$-action.
Warning The lift from $\phi$ to $\hat \phi$ above, hence from the existence of Hamiltonians to an actual choice of Hamiltonians is in general indeed a choice. There may be different choices. In the literature the difference between $\hat \phi$ and $\phi$ (or of their Lie theoretic analogs) is not always clearly made.
By (Atiyah-Bott), the action of a Lie algebra on a symplectic manifold is Hamiltonian if and only if the symplectic form has a (basic, closed) extension to equivariant de Rham cohomology.
projective Hamiltonian action: classical anomaly
A comprehensive account is in (see around section 2.1)
The perspective on Hamiltonian actions in terms of maps to extensions, infinitesimally and integrally, is made explicit in prop. 2.4.10 of
The characterization in equivariant cohomology is due to
Generalization to Hamiltonian actions by a Lie algebroid (instead of just a Lie algebra) is discussed in
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