Some vertical users, e.g. industrial environments, demand customized private 5G networks that address their technological and business needs. These networks present challenges that differ from those of traditional mobile network operators(MNOs), and that should be addressed through innovations in the user, control and management planes. This paper introduces 5G-CLARITY, a 5G-PPP project exploring beyond B5G private networks integrating heterogeneous wireless access including Cellular (5GNR beyond Rel. 16), WiFi, and LiFi. The project targets enhancements to current 5GNR performance including multi-connectivity and indoor positioning accuracy. 5G-CLARITY also develops novel management enablers that allow to operate the private network with a high level intent interface, while being able to natively embed machine learning (ML) functions.