A simple model for a spinning particle is
$$L=m\int dt\left(\dot{x}^{2}-\frac{i}{2}\psi\dot{\psi}\right)$$
with SUSY algebra $\delta x=-i\epsilon\psi$ and $\delta\psi=-\epsilon\dot{x}$, where $\epsilon$ is a Grassmann number.
I understand that the Lagrangian for a bosonic particle coupled with a background gauge field $A$ is
$$L=m\int dt\dot{x}^{2}+i\int dt\dot{x}^{\mu}A_{\mu}.$$
What is the action for a spinning particle coupled with the gauge field?
This e.g. explained in Ref. 1.
The massive spin 1/2 particle without an EM background is described by a Hamiltonian Lagrangian $$ \begin{align}L_H~=~&p_{\mu}\dot{x}^{\mu} +\frac{i}{2}(\psi_{\mu}\dot{\psi}^{\mu}+\psi_5\dot{\psi}^5)\cr &-eH - i \chi Q, \cr H~:=~&\frac{1}{2}(p^2+m^2), \cr Q~:=~&p_{\mu}\psi^{\mu}+m\psi^5 .\end{align}\tag{80}$$ For the massless case $m=0$, see also this related Phys.SE post.
In an EM background, the Hamiltonian $H$ and supercharge $Q$ change to $$ \begin{align} H~:=~&\frac{1}{2}((p-qA)^2+m^2)+\frac{iq}{2}F_{\mu\nu}\psi^{\mu}\psi^{\nu}, \cr Q~:=~&(p_{\mu}-qA_{\mu})\psi^{\mu}+m\psi^5 , \end{align} $$ cf. eq. (122) in Ref. 1.
It would take us too far to try to explain every aspect of the above construction, but let us just briefly mention that it is possible perform an Legendre transformation to a Lagrangian formulation, and to gauge-fix the einbein field $e$, to achieve an action closer to OP's starting point.
References:
F. Bastianelli, Constrained hamiltonian systems and relativistic particles, 2017 lecture notes; Section 2.3 + Chapter 3.
L. Brink, P. Di Vecchia & P. Howe, Nucl. Phys. B118 (1977) 76; eq. (5.8).
O. Corradini & C. Schubert, Spinning Particles in QM & QFT, arXiv:1512.08694; Section 1.5, p. 41.
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$^1$ Conventions: We use the Minkowski sign convention $(-,+,+,+)$ and we work in units where $c=1$.
