Autonomous vehicles (AVs), which can be fully controlled by remote/online operators, could be an extension of ride-sourcing services provided by transportation network companies (TNCs). Meanwhile, substitutive and complementary relationships between ride-sourcing and public transit could also help TNCs increase their profit under certain strategies. Unlike previous studies that generally ignored either AVs or interactions between public transit and ride-sourcing, we introduce AV-involved operational strategies into a multi-modal system. In this paper, we focus on transportation hubs in an urban area and let the TNC assign AVs to these hubs for serving only within that particular hub at a differentiated fare; meanwhile, passengers could choose among AVs, human-driven vehicles (HVs), combined modes with public transit and AVs or HVs, or other travel modes to fulfill their travel needs. We develop a mathematical model to investigate the impacts of such an operational strategy, and formulate an optimisation problem to maximise the TNC's profit and minimise total waiting time simultaneously by adjusting AV fare and fleet size. We further propose a comprehensive modeling framework with the analytical model, optimisation problem, calibration method, and heuristic algorithm, making it a general approach for different real-world scenarios. Following the framework, we conduct a case study based on real-world datasets of public transit and ride-sourcing services in Hangzhou, China. Different market schemes are analyzed and compared with the currently existing situation. The results demonstrate that by adopting this auxiliary-AV-oriented operational strategy, the TNC's profit and public transit ridership can both be increased, and passengers could enjoy a shorter waiting time for HV ride-sourcing trips. Moreover, the TNC is more inclined to allocate more AVs to hubs with large commute needs and uncongested traffic, leading to a high profit for the TNC and a short waiting time for passengers. Societal preferences towards AV trips are also analyzed. The results provide in-depth references for real-world AV-related ride-sourcing operational problems.