Let $W$ be the canonical $\operatorname{Spin}^\mathbb{C}$ spinor bundle on a symplectic 4-manifold $(M, \omega)$, with a compatible $J$ and $g$, so \begin{equation} {W_ + } = {T^{0,0}}{M^*} \oplus {T^{0,2}}{M^*}, \;\;\;... \end{equation} and the determinant line bundle $L = T^{0,2}M^*$.

Now I want to define a connection on $W$. I learn about basic general theory on spinor bundle, and it says a connection on $W$ consists of the Levi-civita connection $\nabla_g$ and a connection $A$ on $L$. In particular, after I have a $\nabla_g$, I still need to **saparately define** the $A$, which does not induce from $\nabla_g$.

But I see in Sergeev's lecture on Vortex and Seiberg-Witten equations, that in the case of $(M, \omega, J, g)$ one can modify $\nabla_g$ by adding a term with Nijenhuis tensor to give a connection to $L = T^{0,2}M^*$, and therefore a full connection on $W$.

Am I right about the general theory that connection $A$ on $L$ is not prescribed by $g$-connection?

In the special case of canonical $\operatorname{Spin}^\mathbb{C}$-bundle on $(M, \omega, J, g)$, is $\nabla_g$ prescribing a canonical connection on $L$, contrasting with the general theory?