= Solution
For the stated <two-dimensional N=(2,2) supersymmetry> conventions, define the <supersymmetric covariant derivatives>
$$
D_\pm=\frac\partial{\partial\theta^\pm}-i\bar\theta^\pm\partial_\pm,
\qquad
\bar D_\pm=-\frac\partial{\partial\bar\theta^\pm}+i\theta^\pm\partial_\pm.
$$
The terms in which a Grassmann derivative hits the explicit Grassmann coordinate cancel the spacetime-derivative terms, while derivatives involving different signs act on independent coordinates. Therefore
$$
\boxed{\{D_+,Q_\pm\}=\{D_+,\bar Q_\pm\}=0}.
$$
The <twisted chiral superfield> constraints $D_-U=\bar D_+U=0$ are solved by the twisted chiral coordinates
$$
\widetilde y^+=x^+-i\theta^+\bar\theta^+,
\qquad
\widetilde y^-=x^-+i\theta^-\bar\theta^-.
$$
The superfield depends only on $(\widetilde y^\pm;\theta^+,\bar\theta^-)$ and has the finite <Grassmann variable> expansion
$$
\boxed{U=u+\theta^+\chi_++\bar\theta^-\widetilde\chi_-+\theta^+\bar\theta^-G},
$$
where every component on the right is evaluated at $\widetilde y$; numerical $\sqrt2$ factors may be absorbed into the component definitions.
For one <chiral superfield> $\Phi$ and one twisted chiral superfield $U$, the most general local two-derivative supersymmetric action is
$$
\boxed{
\begin{aligned}
S={}&\int d^2x\,d^4\theta\,K(\Phi,\bar\Phi,U,\bar U)\\
&+\left[\int d^2x\,d\theta^+d\theta^-\,W(\Phi)+\mathrm{h.c.}\right]\\
&+\left[\int d^2x\,d\theta^+d\bar\theta^-\,\widetilde W(U)+\mathrm{h.c.}\right].
\end{aligned}}
$$
Here $K$ is real, $W$ is a holomorphic <superpotential>, and $\widetilde W$ is a holomorphic <twisted superpotential>. A full superspace integral, a chiral <F-term>, and a twisted F-term each vary by a spacetime or Berezin total derivative, so all three are supersymmetric.
Choose the <Vector R-symmetry> and <Axial R-symmetry> conventions
$$
U(1)_V:\quad\theta^\pm\mapsto e^{i\alpha}\theta^\pm,
\qquad
U(1)_A:\quad\theta^+\mapsto e^{i\beta}\theta^+,
\quad\theta^-\mapsto e^{-i\beta}\theta^-,
$$
with conjugate coordinates transforming oppositely, and assign compatible charges to $\Phi$ and $U$. The D-term is invariant when $K$ is neutral, up to a generalized Kähler transformation. The chiral measure has vector R-charge $-2$ and axial charge zero, whereas the twisted chiral measure has axial R-charge $-2$ and vector charge zero. Hence classical invariance requires
$$
\boxed{
U(1)_V:\ R_V(W)=2,\ R_V(\widetilde W)=0;
\qquad
U(1)_A:\ R_A(W)=0,\ R_A(\widetilde W)=2,
}
$$
together with neutrality of $K$. Equivalently, $W$ and $\widetilde W$ must be quasi-homogeneous with these charges; absent suitable charge assignments, the corresponding superpotential breaks that R-symmetry.
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