Home-based care (HBC) delivers medical and care services in patients' living environments, offering unique opportunities for patient-centered care. However, patient agency is often inadequately represented in shared HBC planning processes. Through 23 multi-stakeholder interviews with HBC patients, healthcare professionals, and care workers, alongside 60 hours of ethnographic observations, we examined how patient agency manifests in HBC and why this representation gap occurs. Our findings reveal that patient agency is not a static individual attribute but a relational capacity shaped through maintaining everyday continuity, mutual recognition from care providers, and engagement with material home environments. Furthermore, we identified that structured documentation systems filter out contextual knowledge, informal communication channels fragment patient voices, and doctor-centered hierarchies position patients as passive recipients. Drawing on these insights, we propose design considerations to bridge this representation gap and to integrate patient agency into shared HBC plans.