  We construct the most general theory of 2D Einstein-dilaton gravity coupled
with $U(1)$ gauge fields that contains all the 2-derivative and the
4-derivative interactions allowed by the diffeomorphism invariance. We
renormalise the 2D action and obtain the vacuum solution as well as the black
hole solution. The vacuum solution in the UV is dominated by Lifshitz$_2$ with
dynamical exponent ($z=\frac{7}{3}$) while on the other hand, the spacetime
curvature diverges as we move towards the deep IR limit. We calculate the
holographic stress tensor and the central charge for the boundary theory. Our
analysis shows that the central charge goes as the inverse power of the
coupling associated to 4-derivative interactions. We also compute the Wald
entropy for 2D black holes and interpret its near horizon divergence in terms
of the density of states. We compare the Wald entropy with the Cardy formula
and obtain the eigen value of Virasoro operator ($L_0$) for our model. Finally,
we explore the near horizon structure of 2D black holes and calculate the
central charge corresponding to the CFT near horizon. We further show that the
near horizon CFT may be recast as a 2D Liouville theory with higher derivative
corrections. We study the Weyl invariance of this generalised Liouville theory
and identify the Weyl anomaly associated to it. We also comment on the
classical vacuum structure of the theory.
