= Solution
Differentiating $X(z)X(w)\sim-(\alpha'/2)\log(z-w)$ gives the contractions needed for <Wick theorem>. The free-boson part of $T$ supplies $1/[2(z-w)^4]$, corresponding to <central charge> one. The cross-contractions between $:\!\partial X\partial X\!:$ and $\partial^2X$ produce the required lower poles but no fourth-order scalar term. Finally,
$$
\partial_z^2\partial_w^2\left[-\frac{\alpha'}2\log(z-w)\right]
=\frac{3\alpha'}{(z-w)^4},
$$
so the product of the two improvement terms contributes $3\alpha'q^2/(z-w)^4$. Matching this with $c/[2(z-w)^4]$ in the <stress-tensor operator-product expansion> gives the <linear dilaton conformal field theory>
$$
\boxed{c=1+6\alpha'q^2}.
$$
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