We have $A = k[[x, y]]$. Write $L = K[x]/(x^p - f)$ for some $f \in A$ and denote $g \in B$ the congruence class of $x$, i.e., the element such that $g^p = f$. By Algebra, Lemma \ref{algebra-lemma-derivative-zero-pth-power} we see that $\text{d}f$ is nonzero in $\Omega_{K/\mathbf{F}_p}$. By More on Algebra, Lemma \ref{more-algebra-lemma-power-series-ring-subfields} there exists a subfield $k^p \subset k' \subset k$ with $p^e = [k : k'] < \infty$ such that $\text{d}f$ is nonzero in $\Omega_{K/K_0}$ where $K_0$ is the fraction field of $A_0 = k'[[x^p, y^p]] \subset A$. Then $$ \Omega_{A/A_0} = A \otimes_k \Omega_{k/k'} \oplus A \text{d}x \oplus A \text{d}y $$ is finite free of rank $e + 2$. Set $\omega_A = \Omega^{e + 2}_{A/A_0}$. Consider the canonical map $$ \text{Tr} : \Omega^{e + 2}_{B/A_0} \longrightarrow \Omega^{e + 2}_{A/A_0} = \omega_A $$ of Lemma \ref{lemma-trace-extends}. By duality this determines a map $$ c : \Omega^{e + 2}_{B/A_0} \to \omega_B = \Hom_A(B, \omega_A) $$ Claim: the cokernel of $c$ is annihilated by a nonzero element of $B$. \medskip\noindent Since $\text{d}f$ is nonzero in $\Omega_{A/A_0}$ we can find $\eta_1, \ldots, \eta_{e + 1} \in \Omega_{A/A_0}$ such that $\theta = \eta_1 \wedge \ldots \wedge \eta_{e + 1} \wedge \text{d}f$ is nonzero in $\omega_A = \Omega^{e + 2}_{A/A_0}$. To prove the claim we will construct elements $\omega_i$ of $\Omega^{e + 2}_{B/A_0}$, $i = 0, \ldots, p - 1$ which are mapped to $\varphi_i \in \omega_B = \Hom_A(B, \omega_A)$ with $\varphi_i(g^j) = \delta_{ij}\theta$ for $j = 0, \ldots, p - 1$. Since $\{1, g, \ldots, g^{p - 1}\}$ is a basis for $L/K$ this proves the claim. We set $\eta = \eta_1 \wedge \ldots \wedge \eta_{e + 1}$ so that $\theta = \eta \wedge \text{d}f$. Set $\omega_i = \eta \wedge g^{p - 1 - i}\text{d}g$. Then by construction we have $$ \varphi_i(g^j) = \text{Tr}(g^j \eta \wedge g^{p - 1 - i}\text{d}g) = \text{Tr}(\eta \wedge g^{p - 1 - i + j}\text{d}g) = \delta_{ij} \theta $$ by the explicit description of the trace map in Lemma \ref{lemma-trace-higher}. \medskip\noindent Let $Y \to \Spec(B)$ be a normal modification. Exactly as in the proof of Lemma \ref{lemma-go-up-separable} we can reduce to the case where $Y$ is finite over a modification $X$ of $\Spec(A)$. By Lemma \ref{lemma-dominate-by-blowing-up-in-points} we may even assume $X \to \Spec(A)$ is the result of a sequence of blowing ups in closed points. Picture: $$ \xymatrix{ Y \ar[d]_\pi \ar[r]_-g & \Spec(B) \ar[d] \\ X \ar[r]^-f & \Spec(A) } $$ We may apply Lemma \ref{lemma-trace-extends} to $\pi$ and we obtain the first arrow in $$ \pi_*(\Omega^{e + 2}_{Y/A_0}) \xrightarrow{\text{Tr}} (\Omega^{e + 2}_{X/A_0})^{**} \xrightarrow{\varphi_X} \omega_X $$ and the second arrow is from Lemma \ref{lemma-compare-differentials-dualizing} (because $f$ is a sequence of blowups in closed points). By duality for the finite morphism $\pi$ this corresponds to a map $$ c_Y : \Omega^{e + 2}_{Y/A_0} \longrightarrow \omega_Y $$ extending the map $c$ above. Hence we see that the image of $\Gamma(Y, \omega_Y) \to \omega_B$ contains the image of $c$. By our claim we see that the cokernel is annihilated by a fixed nonzero element of $B$. We conclude by Lemma \ref{lemma-bound-dualizing-implies-bound}.