Consider a number of chiral superfield $\Phi_i$ with components $A_i$, $\psi_i$, $F_i$, respectively a complex scalar, a 2-component Weyl fermion and an auxiliary complex scalar. The most general supersymmetric renormalizable Lagrangian involving only chiral superfields is given by the $\Phi_i^\dagger \Phi_i$, $\lambda_i \Phi_i$, $m_{ij} \Phi_i \Phi_j$ and $g_{ijk} \Phi_i \Phi_j \Phi_k$ and their hermitian conjugate. Summation over repeated indices is understood. If we integrate appropriately, we get a Lagrangian in real physical space which is (see Wess & Bagger$^1$, eq. 5.11) $$ \mathcal{L} = i\partial_\mu \bar{\psi_i} \bar{\sigma}^\mu \psi_i + A_i^* \Box A_i +F_i^* F_i \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \\ +\left[ m_{ij} \left( A_i F_i -\frac12 \psi_i \psi_j \right) +g_{ijk}(A_i A_j F_k -\psi_i \psi_j A_k) +\lambda_i F_i +\text{h.c.} \right]. $$ We can integrate out the auxiliary field $F_i$. Its equation of motion reads $$ F_k =-\lambda_k^* -m_{ik}^* A_i^* -g_{ijk}^* A_i^* A_j^*, $$ and similarly for the complex conjugate (eq. 5.12). Inserting $F_k$ back into the Lagrangian, we get the mass terms for $A_i$ and $\psi_i$, as well as some $(\psi^2 A)$-like interactions. We also get a potential of the form $\mathcal{V} =F_k^* F_k$. If we write $\mathcal{V}$ using the equations of motion for $F_k, F_k^*$ we'll have terms of the form $$\tag{1}\label{eq1} \mathcal{L} \supset \lambda_k m_{ij}^* A_k^* +\lambda_k^* m_{ij} A_k +\lambda_k g_{ijk}^* A_i^* A_j^* +\lambda_k^* g_{ijk} A_i A_j. $$ It's maybe clearer from eq. 8.33 of Müller-Kirsten & Wiedemann$^2$, where the notation is swapped: they use $g$ for $\lambda$ and vice versa. Eq. 8.33 in particular is the simplified case of a single superfield $\Phi$ (so no $i,j,k$ indices). They furthermore assume that $g,\lambda,m$ are real. Also note that they are forgetting the mass term for $A$: $m^2 |A|^2$. Eq. \eqref{eq1} becomes (in Wess & Bagger notation) $$ \mathcal{L} \supset \lambda m (A^* +A) +\lambda g ( A^{*2} +A^2). $$ Note that since $A$ is complex, the last piece is not a mass term.
My question is: how should I interpret the Lagrangian terms with only one field $A$ or $A^2$? I couldn't find any comment anywhere, neither in books nor in lecture notes. Are these terms related to tadpoles and wave-function renormalization (Do interactions with 2 legs (or one leg) exist in Standard Model?)? Do I care about these terms if I want to compute the $2\to2$ scattering amplitude for the process $AA\to AA$?
$^1$ https://press.princeton.edu/books/paperback/9780691025308/supersymmetry-and-supergravity
$^2$ https://www.worldscientific.com/worldscibooks/10.1142/7594#t=aboutBook
