  We develop a quenched thermodynamic formalism for random dynamical systems
generated by countably branched, piecewise-monotone mappings of the interval
that satisfy a random covering condition. Given a random contracting potential
$\varphi$ (in the sense of Liverani-Saussol-Vaienti), we prove there exists a
unique random conformal measure $\nu_\varphi$ and unique random equilibrium
state $\mu_\varphi$. Further, we prove quasi-compactness of the associated
transfer operator cocycle and exponential decay of correlations for
$\mu_\varphi$. Our random driving is generated by an invertible, ergodic,
measure-preserving transformation $\sigma$ on a probability space
$(\Omega,\mathscr{F},m)$; for each $\omega\in\Omega$ we associate a
piecewise-monotone, surjective map $T_\omega:I\to I$. We consider general
potentials $\varphi_\omega:I\to\mathbb R\cup\{-\infty\}$ such that the weight
function $g_\omega=e^{\varphi_\omega}$ is of bounded variation. We provide
several examples of our general theory. In particular, our results apply to
linear and non-linear systems including random $\beta$-transformations,
randomly translated random $\beta$-transformations, random Gauss-Renyi maps,
random non-uniformly expanding maps such as intermittent maps and maps with
contracting branches, and a large class of random Lasota-Yorke maps.
