The non-noble metal (Fe, Co, Ni, etc.) catalysts possess promising potential to replace noble metals (e.g., Pt, Ru, Ir, etc.) as catalysts for oxygen electrocatalysis. Up to now, various mono- and dual-single-atom catalysts have been fabricated, though it is still challenging to synthesise ternary single-atom catalysts due to the difference of interaction forces between different metal ions (Fe, Co, Ni, etc.) and ligands. Here, we report a Fe-Co-Ni ternary single-atom catalyst (FeCoNi-N_x) derived from a zeolitic imidazolate frameworks (ZIF) precursor as an efficient oxygen electrocatalyst, and an optimised flexible casting-drying polyvinyl alcohol (CD-PVA) film as a quasi-solid electrolyte host, for high-efficiency solid-state Zn-air batteries. The aberration-corrected HAADF-STEM and EELS spectrum confirm the co-existence of Fe, Co and Ni single atoms in the FeCoNi-N_x catalyst, and the electrochemical, mechanical, and durability tests prove the superiority of the CD-PVA film. As a result, the FeCoNi-N_x-based rechargeable Zn-air battery delivers superior specific capacity (846.8 mAh·g_Zn^–1) and power density (135 mW·cm^–2) in aqueous electrolyte, as well as an over 60 mW·cm^–2 power density in quasi-solid electrolyte. As a result, the quasi-solid-state Zn-air battery with a small area of only 2 cm² is able to charge a mobile phone, which outperforms all the reported devices to date.