We construct a measure (of finite total mass) on the space of metrics of a fixed 1-dimensional complex manifold (of genus greater than or equal to 2) which corresponds in the physics literature to a path integral based on a Gibbs type measure with energy given by the sum of the Liouville action and the Mabuchi K-energy. To the best of our knowledge, this is the first rigorous construction of such an object and this is done by means of probabilistic tools. Both functionals (the Liouville functional and the K-energy functional) play an important role, respectively, in Riemannian geometry (in the case of surfaces) and Kähler geometry. As an output, we obtain a measure whose Weyl anomaly displays the standard Liouville anomaly plus an additional K-energy. Motivations come from theoretical physics where these type of path integrals arise as models for fluctuating metrics on surfaces when coupling certain nonconformal matter fields (mathematically noncritical statistical physics models) to quantum gravity as advocated by A. Bilal, F. Ferrari, S. Klevtsov, and S. Zelditch. On the discrete side, our measure is expected to describe the scaling limit of large planar maps decorated by some noncritical statistical physics models. Interestingly, our computations show that quantum corrections perturb the classical Mabuchi K-energy and produce a quantum Mabuchi K-energy: these type of corrections are reminiscent of the quantum Liouville theory. Our probabilistic construction relies on a variant of the theory of Gaussian multiplicative chaos (GMC) and derivative GMC (DGMC for short). The technical backbone of our construction consists in two estimates on (derivative and standard) GMC which are of independent interest in probability theory. First, we show that these DGMC random variables possess negative exponential moments, and second, we derive optimal small deviations estimates for the GMC associated to a recentered Gaussian free field (GFF).