= Solution
The <Kähler potential> $K(\Phi,\Phi^\dagger)$ is a real function integrated over all four fermionic coordinates, $\int d^4\theta\,K$. Its complex Hessian gives the scalar <Kähler metric> and therefore the kinetic terms. The <superpotential> $W(\Phi)$ is holomorphic and is integrated over chiral superspace, $\int d^2\theta\,W+\mathrm{h.c.}$; its derivatives determine Yukawa couplings and the <F-term scalar potential>.
For the canonical <Kähler potential> $K=\Phi^\dagger\Phi$, extracting the $\theta^2\bar\theta^2$ component of the supplied <chiral-superfield component expansion> and integrating by parts gives the bosonic action
$$
\int d^4x\,d^4\theta\,\Phi^\dagger\Phi
=\int d^4x\left(-\partial_\mu\phi^*\partial^\mu\phi+F^*F\right),
$$
in the mostly-plus convention. Extracting the $\theta^2$ component of a holomorphic function gives
$$
\int d^4x\,d^2\theta\,W(\Phi)
=\int d^4x\left(FW'(\phi)-\frac12W''(\phi)\psi\psi\right),
$$
so its bosonic term is $FW'(\phi)$, with the Hermitian conjugate understood in a real action. For several fields, eliminating each algebraic <auxiliary field> by $F_i^*=-W_i$ produces $V_F=\sum_i|W_i|^2$.
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