Supersymmetry only plays a minor role here, so we will suppress it in this answer. The Faddeev-Popov gauge-fixing procedure and the background field method $$ A_{\mu}^a~=~\underbrace{\overline{A}_{\mu}^a}_{\text{background}} +\underbrace{{\cal A}_{\mu}^a}_{\text{quant. fluct.}} \tag{A}$$ are e.g. explained for Yang-Mills theory in Ref. 1, chapter 71 & 78, respectively. Using the notation & conventions of Ref.1, the resulting Lagrangian density (without matter fields) becomes $${\cal L}~=~-\frac{1}{4}F_{\mu\nu}^a F^{\mu\nu a} + \underbrace{\overline{c}^a\overline{D}^{\mu} D_{\mu}c^a}_{\text{Faddeev-Popov term}} - \underbrace{\frac{1}{2\xi} G^aG^a}_{\text{gauge-fixing term}},\tag{B}$$ with gauge-covariant derivative $$ D_{\mu}~=~\partial_{\mu}-igA_{\mu},\qquad \overline{D}_{\mu}~=~\partial_{\mu}-ig\overline{A}_{\mu},\tag{C}$$ 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:
- M. Srednicki, QFT, 2007; Chapter 71 & 78. A prepublication draft PDF file is available here.