= Solution
The first term of the action is the <free boson conformal field theory>. For the curvature coupling, insert the supplied <first variation> of $\sqrt gR^{(2)}$, integrate by parts twice and discard the boundary term. Its metric variation is the stress-tensor improvement
$$
-q\left(\nabla_\alpha\nabla_\beta X-g_{\alpha\beta}\nabla^2X\right)
$$
in the conventions of the question. In a locally flat complex coordinate, the holomorphic component of the complete <stress-energy tensor> is therefore
$$
T(z)=-\frac1{\alpha'}:\!\partial X\partial X\!:(z)-q:\!\partial^2X\!:(z),
$$
as required. Thus the coupling $q\int\sqrt g\,XR^{(2)}$ makes $X$ a <background-charge scalar field>, equivalently a worldsheet coordinate in a <linear dilaton conformal field theory>.
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