High-average-power extreme ultraviolet (EUV) sources are essential for large-scale nanoscale chip manufacturing, yet commercially available laser-produced plasma sources face challenges in scaling to the kilowatt level. We propose a novel scheme that combines the high repetition rate of a diffraction-limited storage ring with a regenerative amplifier free-electron laser (RAFEL) employing a transverse gradient undulator (TGU). By introducing dispersion in the storage ring, electrons of different energies are directed into corresponding magnetic field strengths of the TGU, thereby satisfying the resonance condition under a large energy spread and increasing the FEL gain. Simulations show that at equilibrium, the average EUV power exceeds 1 kW, with an output pulse energy reaching ∼2.86 μJ, while the energy spread stabilizes at ∼0.45%. These results demonstrate the feasibility of ring-based RAFEL with TGU as a promising route toward kilowatt-level EUV sources.