we collect a number of background statements concerning solutions of perturbed Cauchy-Riemann equations. For the rest of the paper, fix a real number $p\!>\!2$. If $\Si$ is a $2$-dimensional manifold, this condition implies that any $L^p_1$-map $\Si\!\lra\!\R$ is continuous and in particular has a well-defined value at each point.

Nodal Riemann surfaces {#NRS_subs}
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Let $(E,\fI)\!\lra\!\Si$ be an $L^p_1$-complex vector bundle over a smooth Riemann surface, i.e. a one-dimensional complex manifold. If $z\!\in\!\Si$ and $$A_z\in\Hom_{\R}(E_z,T_z^*\Si^{0,1}\!\otimes_{\C}\!E_z),$$ we define $$\begin{gathered}
A_z^*\in\Hom_{\R}(T_z^*\Si^{1,0}\!\otimes_{\C}\!E_z^*,T_z^*\Si^{1,1}\!\otimes_{\C}\!E_z^*)
\qquad\hbox{by}\\
\Re\big(v\wedge (A_z^*w)\big)= \Re\big((A_zv)\wedge w\big)
\in \La_{\R}^2(T_z^*\Si)
\qquad\forall\, v\!\in\!E_z,\,w\!\in\!T_z^*\Si^{1,0}\!\otimes_{\C}\!E_z^*\,.\end{gathered}$$ Since $\La_{\R}^2(T_z^*\Si)$ is one-dimensional, $A_z^*$ is well-defined. If $$A\in L^p\big(\Si;\Hom_{\R}(E,T^*\Si^{0,1}\!\otimes_{\C}\!E)\big),$$ this construction gives rise to an element $$\begin{gathered}
A^*\in L^p\big(\Si;\Hom_{\R}(T^*\Si^{1,0}\!\otimes_{\C}\!E^*,
T^*\Si^{1,1}\!\otimes_{\C}\!E^*)\big) \qquad\hbox{s.t.} \notag\\
\label{Aadj_e}
\bllrr{\xi,A^*\eta}\equiv
\Re\Big(\int_{\Si}\xi\wedge (A^*\eta)\Big)
=\Re\Big(\int_{\Si}(A\xi)\wedge\eta\Big)
\equiv \bllrr{A\xi,\eta}\end{gathered}$$ for all $\xi\!\in\!L^p_1(\Si;E)$ and $\eta\!\in\!L^p_1(\Si;T^*\Si^{1,0}\!\otimes\!E^*)$.\
Let $E\!\lra\!\Si$ be as above. If $S$ is a finite subset of $\Si$, denote by $$L_k^p\big(\Si;E(S)\big)\subset L^p_{k,loc}(\Si\!-\!S;E)$$ the subspace of sections $\eta$ of $E$ such that for every $z_0\!\in\!S$ there exist a neighborhood $U$ of $z_0$ in $\Si$ and a coordinate $w\!:U\!\lra\!\C$ such that $$w(z_0)=0 \qquad\hbox{and}\qquad
w\cdot\eta|_U\in L_k^p(U;E).$$ If $k\!\ge\!1$, an element $\eta$ of $L_k^p(\Si;T^*\Si^{1,0}\!\otimes_{\C}\!E(S))$ has a well-defined residue at $z_0\!\in\!S$ given by $$\Res_{z=z_0}\eta=\xi(z_0)\in E_{z_0} \qquad\hbox{if}\qquad
\eta(z)=\frac{dw}{w(z)}\otimes\xi(z)~~\forall~z\!\in\!U,~
\xi\in L_1^p(U;E).\footnotemark$$ If $\vr$ is a function assigning to each 