= Solution
Use the canonical <Kähler potential> and normalize <superspace integration> by $\int d^2\theta\,\theta^2=1$. The <superspace> <Lagrangian density> is
$$
\boxed{\mathcal L=\int d^4\theta\,\Phi^\dagger\Phi+
\left[\int d^2\theta\left(\frac m2\Phi^2+\frac g3\Phi^3\right)+\mathrm{h.c.}\right].}
$$
The full <superspace integration> produces a <D-term>, while the <chiral superfield> integral produces an <F-term>. Work in chiral coordinates for the <chiral-superfield component expansion>, so the omitted spacetime-derivative terms do not enter the extraction of $F$.
Set $\delta\Phi=\sqrt2\theta\psi+\theta^2F$. A <Taylor expansion> of the <superpotential> gives
$$
W(\varphi+\delta\Phi)=W(\varphi)+W'(\varphi)\delta\Phi+
\frac12W''(\varphi)(\delta\Phi)^2.
$$
Higher terms vanish because there are only two <Grassmann variables> $\theta^\alpha$. With $\theta\psi=\theta^\alpha\psi_\alpha$ and $\psi\psi=\psi^\alpha\psi_\alpha$, their anticommutation gives $(\theta\psi)^2=-\tfrac12\theta^2\psi\psi$. Thus
$$
\left.W(\Phi)\right|_{\theta^2}=F W'(\varphi)-\frac12W''(\varphi)\psi\psi,
\qquad W'=m\varphi+g\varphi^2,\qquad W''=m+2g\varphi.
$$
The canonical <Kähler potential> contributes $F^*F$. Consequently the part of the <Lagrangian density> containing the <auxiliary field> is
$$
\boxed{\mathcal L_F=F^*F+F(m\varphi+g\varphi^2)
+F^*(m^*\varphi^*+g^*\varphi^{*2}).}
$$
The <fermion> terms coming from the <superpotential> are
$$
\mathcal L_{\mathrm{mass+Yukawa}}=-\frac12(m+2g\varphi)\psi\psi
-\frac12(m^*+2g^*\varphi^*)\bar\psi\bar\psi.
$$
The <Weyl spinor> kinetic term and the <complex scalar field> kinetic term are canonical because $K=\Phi^\dagger\Phi$.
Treat $F$ and $F^*$ as independent variables in their algebraic <Euler-Lagrange equations>. They give
$$
F^*=-W'(\varphi),\qquad F=-\overline{W'(\varphi)}.
$$
Equivalently,
$$
\mathcal L_F=\bigl|F+\overline{W'(\varphi)}\bigr|^2-|W'(\varphi)|^2.
$$
After eliminating the <auxiliary field>, the <Lagrangian density> contains $-V$, so the <F-term scalar potential> is
$$
\boxed{V(\varphi)=|m\varphi+g\varphi^2|^2
=|m|^2|\varphi|^2+m^*g\varphi^*\varphi^2
+mg^*\varphi\varphi^{*2}+|g|^2|\varphi|^4\geq0.}
$$
It is nonnegative for arbitrary complex $m,g,\varphi$, since it is a <modulus> squared. A <supersymmetric vacuum> satisfies <F-flatness>, so its zeros are:
$$
\varphi=0,\qquad \varphi=-\frac m g\quad(g\ne0).
$$
If $m=0$ and $g\ne0$ these coincide; if $g=0$ and $m\ne0$ only the origin remains; if $m=g=0$ every constant $\varphi$ is a <supersymmetric vacuum>.
To compare masses, expand around a <supersymmetric vacuum> $v$ satisfying $W'(v)=0$, writing $\varphi=v+\eta$. With $M=W''(v)=m+2gv$,
$$
W'(v+\eta)=M\eta+g\eta^2,\qquad
V=|M|^2|\eta|^2+\text{terms of degree at least three},
\qquad \mathcal L_{\mathrm{fermion\ mass}}=-\frac12M\psi\psi+\mathrm{h.c.}
$$
Writing $\eta=(A+iB)/\sqrt2$ gives the <scalar field> mass term $-\tfrac12|M|^2(A^2+B^2)$; the <Weyl spinor> has physical mass $|M|$. Therefore the <supersymmetric mass degeneracy> is
$$
\boxed{m_A^2=m_B^2=|W''(v)|^2,\qquad m_\psi=|W''(v)|.}
$$
In particular, at $v=0$ the <boson> and <fermion> masses are both $|m|$. At the second <supersymmetric vacuum> $v=-m/g$ the sign of $M$ reverses, but the physical mass is again $|m|$. Equality here concerns fluctuations about a <supersymmetric vacuum>, not arbitrary backgrounds with $W'(v)\ne0$.
In the convention $V\supset\lambda|\varphi|^4$ and $\mathcal L_{\mathrm{Yukawa}}=-y\varphi\psi\psi+\mathrm{h.c.}$, the same <superpotential> coefficient gives $\lambda=|g|^2$ and $y=g$. The <supersymmetric relation between quartic and Yukawa couplings> is therefore
$$
\boxed{\lambda=|y|^2.}
$$
If instead the <Yukawa interaction> is written $-\tfrac12y_3\varphi\psi\psi+\mathrm{h.c.}$, then $y_3=2g$ and the same physical relation is $\lambda=|y_3|^2/4$. The numerical factor depends on the definition of the <Yukawa coupling>.
The perturbative <non-renormalization theorem> applies to the local <holomorphic> <superpotential> in the <Wilsonian effective action>, using <regularization in quantum field theory> that preserves <supersymmetry>. There are two complementary ways to see why it applies here.
First, in <supergraph> calculations in <perturbative quantum field theory>, the algebra of <supersymmetric covariant derivatives> puts any candidate local correction to the <superpotential> into a full $d^4\theta$ <superspace integration>. It therefore corrects a <D-term>, such as the <Kähler potential>, rather than a local chiral $d^2\theta$ <F-term>. Turning such a contribution into an apparent chiral integral would require nonlocal inverse spacetime derivatives, schematically $1/\Box$. These are excluded from the local <Wilsonian effective action> with a fixed nonzero momentum cutoff. This gives the diagrammatic <non-renormalization theorem>.
Second, the <holomorphy argument for superpotential non-renormalization> treats $m$ and $g$ as background chiral <spurions>, so the <Wilsonian effective action> <superpotential> is a <holomorphic function> of $\Phi,m,g$, with no dependence on $m^*$ or $g^*$. Assign an ordinary $U(1)$ spurion charge and an <R-charge> as follows:
$$
\begin{array}{c|cccc}
&\Phi&m&g&\theta\\\hline
U(1)&1&-2&-3&0\\
R&1&0&-1&1
\end{array}
$$
The <superpotential> has ordinary charge zero and <R-charge> two, since $d^2\theta$ has <R-charge> minus two. A prospective <holomorphic> monomial $m^a g^b\Phi^c$ must therefore satisfy
$$
c-2a-3b=0,\qquad c-b=2,
\qquad\text{hence}\qquad a=3-c,\quad b=c-2.
$$
For perturbative corrections, regularity at $m=0$ and $g=0$ requires nonnegative powers of the couplings. Regularity in $m$ is justified by keeping a nonzero Wilsonian momentum cutoff and retaining $\Phi$ as a light field. Hence only $c=2$ and $c=3$ survive: the existing $m\Phi^2$ and $g\Phi^3$ structures. The quadratic coefficient is fixed by the free theory at $g=0$, where there are no interaction loops. The cubic coefficient is fixed at first order in $g$, where a connected three-field interaction has only its <tree-level Feynman diagram>; loop corrections would require further powers of the interaction couplings, which the <holomorphy argument for superpotential non-renormalization> charge constraints forbid. Thus
$$
\boxed{W_{\mathrm{Wilsonian}}(\Phi)=\frac m2\Phi^2+\frac g3\Phi^3
\quad\text{to all perturbative orders in holomorphic variables}.}
$$
A field-independent constant is physically irrelevant in this rigid model; the spurion charges would in any case require $m^3/g^2$, which is not regular at $g=0$.
The <Kähler potential> can still receive <wave-function renormalization>. For example, $K=Z\Phi^\dagger\Phi+\cdots$ gives the <canonical field normalization> $\Phi_c=Z^{1/2}\Phi$, and consequently
$$
m_c=\frac m Z,\qquad g_c=\frac g{Z^{3/2}}.
$$
These canonically normalized parameters can run with the <renormalization scale>; this does not contradict the <holomorphic> <non-renormalization theorem>. When massless modes are included in the full <quantum effective action>, nonlocal terms associated with <infrared divergences> can also imitate an <F-term>. Distinguishing that action from the local <Wilsonian effective action> is part of the theorem's convention.
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