Coal fly ash (CFA) is a waste that forms via coal combustion in thermal power stations. CFA consists of numerous components, whose recovery can address environmental and resource concerns associated with sustainable development. Most of the alumina (Al2O3) and rare-earth elements (REEs) in CFA are contained in the amorphous glassy mass and in the refractory mullite phase (3Al2O3·SiO2), which can be dissolved only using high-pressure acid leaching (HPAL). In this paper, the method of preactivation of CFA by treatment with a highly concentrated NaOH solution is used to increase the efficiency of Al and Sc extraction during HPAL. This method allows for the elimination of an inert aluminosilicate layer from the surface of mullite, transferring the REEs into an acid-soluble form. The Al and Sc extraction can reach 80% after HCl HPAL at T = 170 °C and a 90 min duration. According to the kinetic data, the dissolution of Al follows the surface chemical reaction and intraparticle diffusion shrinking core models in the initial and later stages of leaching, respectively. A high activation energy of 52.78 kJ mol−1 was observed at low temperatures, and a change in the mechanism occurred after 170 °C when the activation energy decreased to 26.34 kJ mol–1. The obtained activation energy value of 33.51 kJ mol−1 for Sc leaching indicates that diffusion has a strong influence at all studied temperatures. The residue was analysed by SEM-EDX, XRF, BET, and XRD methods in order to understand the mechanism of DCFA HPAL process.