We develop a unified perturbative framework for energy transport along a chiral quantum Hall (QH) edge coupled to a disordered, compressible strip. Treating the strip as a generic linear-response environment characterized by its retarded susceptibility ${\chi }_q^R(k,\omega )$, we obtain leading corrections to both the heat flux carried by the edge plasmon and to its spectrum. Two generic regimes emerge: (i) a gapped, local dielectric response with finite-range coupling, producing a negative correction to the quantized heat flux that scales as $T^4$ at low temperatures together with a convex cubic shift of the plasmon dispersion, and (ii) a hydrodynamic (diffusive) response with relaxation, yielding a crossover from $T^4$ to $T^{3/2}$ scaling and a change of sign in the correction. We further introduce a microscopic dipolar model in which the edge couples electrostatically to localized dipole moments inside a wide compressible strip. This long-range interaction amplifies the nonlocal dielectric back-action and generates new suppression laws, $T^3$, or even $T^2$ for smooth disorder profiles, together with a universal ratio connecting spectral curvature to thermal response. Across all regimes, the total heat flux remains quantized: The apparent deficit of the plasmon contribution reflects a reversible heat drag into the compressible strip rather than a breakdown of quantization. The framework thus provides a coherent and quantitatively plausible explanation of the “missing heat flux” anomaly and unifies the thermal and spectral signatures of QH edge dynamics.

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• Ballistic transport
• Capacitance
• Carrier dynamics
• Charge
• Charge density waves
• Charge dynamics
• Dielectric properties
• Diffusion
• Dissipative dynamics
• Edge states
• Electric polarization
• Electron thermal conductivity
• Hall effect
• Integer quantum Hall effect
• Landau levels
• Plasma oscillations
• Quantum transport
• Temperature