Supersymmetry only plays a minor role here. The [Faddeev-Popov](https://en.wikipedia.org/wiki/Faddeev%E2%80%93Popov_ghost) gauge-fixing procedure and the [background field method](https://en.wikipedia.org/wiki/Background_field_method) 
$$ A_{\mu}^a~=~\overline{A}_{\mu}^a+{\cal A}_{\mu}^a \tag{A}$$
are e.g. explained for [Yang-Mills theory](https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory) in Ref. 1, chapter 71 & 78, respectively. Using the notation & conventions of Ref.1, the Lagrangian density (without matter fields) becomes
$${\cal L}~=~-\frac{1}{4}F_{\mu\nu}^a F^{\mu\nu a} + \overline{c}^a\overline{D}^{\mu}  \left(\overline{D}_{\mu}c^a-ig[{\cal A}_{\mu}, c]^a\right)  - \frac{1}{2\xi} G^aG^a,\tag{B}$$
with gauge-covariant derivative
$$ D_{\mu}~=~\partial_{\mu}-igA_{\mu},\tag{C}$$
and $R_{\xi}$ gauge-fixing function
$$G^a~:=~\overline{D}^{\mu}{\cal A}_{\mu}^a.\tag{D}$$
We leave it to the reader to translate the Lagrangian density (B) into the notation of the Becker sisters.


References: 

1. M. Srednicki, _QFT,_ 2007; Chapter 71 & 78. A prepublication draft PDF file is available [here](http://web.physics.ucsb.edu/~mark/qft.html).