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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 $φ$ (in the sense of Liverani-Saussol-Vaienti), we prove there exists a unique random conformal measure $ν_φ$ and unique random equilibrium state $μ_φ$. Further, we prove quasi-compactness of the associated transfer operator cocycle and exponential decay of correlations for $μ_φ$. Our random driving is generated by an invertible, ergodic, measure-preserving transformation $σ$ on a probability space $(Ω,\mathscr{F},m)$; for each $ω\inΩ$ we associate a piecewise-monotone, surjective map $T_ω:I\to I$. We consider general potentials $φ_ω:I\to\mathbb R\cup\{-\infty\}$ such that the weight function $g_ω=e^{φ_ω}$ 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 $β$-transformations, randomly translated random $β$-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.