= Solution
Use the canonical <chiral-superfield component expansion> $\Phi=\phi+\sqrt2\theta\psi+\theta\theta F$ in chiral coordinates, with $\psi\psi=\psi^\alpha\psi_\alpha$. Multiplication of the <Grassmann variables> gives
$$
[\Phi^2]_{\theta\theta}=2\phi F-\psi\psi,\qquad [\Phi^3]_{\theta\theta}=3\phi^2F-3\phi\psi\psi.
$$
For example, the two linear-in-$\theta$ factors in $\Phi^2$ contribute $-\theta\theta\psi\psi$, while each of the three choices of the remaining scalar in $\Phi^3$ gives the same contribution. Thus the required <F-term> is
$$
\boxed{[W]_{\theta\theta}=(m\phi+g\phi^2)F-\frac12(m+2g\phi)\psi\psi.}
$$
Adding its Hermitian conjugate and the canonical $F^*F$ term eliminates the <auxiliary field> through $F=-\overline{m\phi+g\phi^2}$ and yields the <F-term scalar potential>
$$
V=|m\phi+g\phi^2|^2=|\phi|^2|m+g\phi|^2.
$$
It is nonnegative, and for $g\ne0$ its absolute minima are \b[$\phi=0$ and $\phi=-m/g$], each with zero energy. They coincide when $m=0$. If $g=0,m\ne0$, only $\phi=0$ is a minimum; if $g=m=0$, every $\phi$ is a minimum. These exhaust the local minima as well: at any other <stationary point> $V_{\phi}=W''\overline{W'}=0$ forces $W''=0$; for $mg\ne0$ this is $\phi=-m/(2g)$, whose quadratic variation has opposite signs in two real directions and is a saddle.
Expand around $\phi=0$, writing $\phi=h=(h_1+ih_2)/\sqrt2$. The <scalar potential> and fermionic terms are
$$
\begin{aligned}
V&=|m|^2|h|^2+m^*g\,h^2h^*+mg^*\,h(h^*)^2+|g|^2|h|^4,\\
\mathcal L_{\mathrm{fermion}}&\supset-\frac12m\psi\psi-g h\psi\psi+\mathrm{h.c.}
\end{aligned}
$$
The canonical <kinetic term> is $\frac12[(\partial h_1)^2+(\partial h_2)^2]$. Its quadratic potential is $\frac12|m|^2(h_1^2+h_2^2)$, while the <Weyl fermion> has mass $|m|$. Hence
$$
\boxed{m_{h_1}=m_{h_2}=m_\psi=|m|.}
$$
At the other minimum, $W''=-m$, so the same mass equality holds. The quartic and <Yukawa interaction> are both fixed by $g$: defining $V\supset\lambda|h|^4$ and $\mathcal L\supset-yh\psi\psi+\mathrm{h.c.}$ gives
$$
\boxed{y=g,\qquad\lambda=|g|^2=|y|^2.}
$$
This is the <supersymmetric relation between quartic and Yukawa couplings>. The printed equality of the two couplings refers to their common supersymmetric origin; literal equality of the conventional coefficients $\lambda$ and $y$ is not true for arbitrary $g$. If instead the Yukawa term is defined as $-\frac12yh\psi\psi$, then $y=2g$ and $\lambda=|y|^2/4$.
To establish perturbative <superpotential non-renormalization>, use a <Wilsonian effective action> with a fixed positive infrared cutoff, so integrating out modes does not introduce massless infrared singularities. Promote $m,g$ to nondynamical <chiral superfields>, or <spurions>. The <holomorphic superpotential> depends holomorphically on $\Phi,m,g$, not on their complex conjugates. Assign an ordinary $U(1)$ charge and an <R-charge> by
$$
\begin{array}{c|ccc}
&\Phi&m&g\\\hline
q&1&-2&-3\\
R&1&0&-1
\end{array}.
$$
The <superspace> coordinate has $R(\theta)=1$, so $d^2\theta$ has charge $-2$ and $W$ must have <R-charge> $2$. Both original monomials satisfy the two selection rules. A <holomorphic> perturbative monomial $m^ag^b\Phi^c$ must obey
$$
-2a-3b+c=0,\qquad-b+c=2,
$$
which gives $c=b+2$ and $a=1-b$. Equivalently,
$$
W_{\mathrm{eff}}=m\Phi^2f(g\Phi/m),\qquad W_{\mathrm{eff}}^{\mathrm{pert}}=\sum_{b\geq0}c_b\,g^b m^{1-b}\Phi^{b+2}.
$$
The nonnegative integer $b$ follows from perturbative analyticity in $g$. Regularity as $m\to0$ at the fixed Wilsonian cutoff excludes $b\geq2$, which would require negative powers of $m$. The remaining quadratic coefficient is fixed to $c_0=1/2$ by the free theory at $g=0$. The coefficient $c_1$ is $1/3$: a three-point contribution at first order in $g$ has just one cubic vertex, uses all three legs externally, and cannot contain a loop. A loop correction would require additional interactions, whereas the selection rules and <holomorphy> have already excluded the corresponding monomials. Therefore
$$
\boxed{W_{\mathrm{eff}}^{\mathrm{pert}}=\frac12m\Phi^2+\frac13g\Phi^3.}
$$
This proof concerns the local Wilsonian <superpotential>. The <Kähler potential> can acquire a wave-function factor $Z$, and canonical normalization changes the couplings to $m_c=m/Z$ and $g_c=g/Z^{3/2}$. Such running, and nonlocal infrared terms in a one-particle-irreducible action, do not contradict the <holomorphic> result.
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