= Solution
Extract a mode using
$$
G_m=\oint_0\frac{dz}{2\pi i}\,z^{m+1/2}G(z),
\qquad
L_k=\oint_0\frac{dw}{2\pi i}\,w^{k+1}T(w).
$$
In the radially ordered double contour for the anticommutator, the simple pole $2T(w)/(z-w)$ gives $2L_{m+n}$. Expanding $z^{m+1/2}$ about $w$, the third-order pole contributes one half of its second derivative,
$$
\frac12\left(m+\frac12\right)\left(m-\frac12\right)
w^{m-3/2}.
$$
The remaining contour is nonzero only for $m+n=0$. Restoring the general leading OPE coefficient $2c/3$ gives
$$
\boxed{
\{G_m,G_n\}=2L_{m+n}
+\frac c{12}(4m^2-1)\delta_{m,-n}}.
$$
This is the fermionic relation in the <N=1 super-Virasoro algebra>.
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