Khvicha came through Dinamo Tbilisi's youth setup and made his senior debut, but contractual disputes prompted a move to the smaller club FC Rustavi. A proposed transfer to Bayern Munich collapsed, so shortly after his 18th birthday he joined Lokomotiv Moscow on loan and, in summer 2019—by then a Georgia international—signed permanently for Rubin Kazan.
During three seasons in Kazan, he matured into one of the Russian Premier League's most exciting talents, drawing fresh interest from Bayern in 2021. "I am convinced he will end up at Real Madrid or Bayern," his agent Nobel Dzhugueli stated at the time. The move fell through, and he remained in Kazan until Russia's invasion of Ukraine in March 2022. He then spent a brief spell on loan with Georgian top-flight side Dinamo Batumi, a spell that underlined his class above the domestic league.
In summer 2022, SSC Napoli paid €11.5m for his services; at the foot of Vesuvius he quickly earned the nickname 'Kvaradona' and helped the club secure the Scudetto. In January 2025 he moved to PSG for €80m; six months later he lifted the Champions League trophy in Munich. Widely regarded as one of world football's brightest talents, he then delivered a brace in Napoli's dramatic 5-4 first-leg win over Bayern.