Vertical users demand customized private 5G networks that address their specific technological and business needs. However, these networks present specific challenges that differ from those of traditional MNOs, which should be addressed through innovations in the user, control and management planes. This paper introduces 5G-CLARITY, a 5GPPP project investigating private network architectures beyond 3GPP Release 16 that integrate 5GNR with IEEE 802.11 WiFi and LiFi, while supporting multi-connectivity and enhanced localization. 5GCLARITY also develops novel management enablers that allow private network users to operate the network with a high level intent interface, while being able to natively embed Machine Learning (ML) functions.