Let $(M, \omega)$ be a symplectic manifold and $L \subseteq M$ - a Lagrangian submanifold. I am trying to understand under what circumstances the Maslov homomorphism $I_{\mu, L} \colon \pi_2(M, L) \to \mathbb{Z}$ is in fact induced by an element $\mu_L \in H^2(M, L;\mathbb{Z})$. I am encountering a few related issues:

If I have a map $f \colon (S, \partial S) \to (M, L)$ from an oriented surface $S$ with non-empty boundary, I can symplectically trivialise $(f^*TM, \omega)$ and calculate the Maslov index of $f\vert_{\partial S}^*TL$ using this trivialisation and the natural orientations of the different boundary components of $S$. But since $Sp(2n)$ is not simply connected, this might depend on the choice of trivialisation if $S$ has non-zero genus.

A related question is when is a class $a \in H_2(M, L; \mathbb{Z})$ representable by a surface as above?

Even if the Hurewicz homomorphism $h\colon \pi_2(M, L) \to H_2(M, L;\mathbb{Z})$ is surjective, so I can represent every relative homology class by a disc, do I have that the Maslov index of a disc $a \in \pi_2(M, L)$ depends in fact only on $h(a)$?