Electronic health records (EHRs) provide longitudinal evidence for clinical prediction, but EHR data are sparse, incomplete, and heterogeneous, which can limit robustness. Medical knowledge graphs (MKGs) have therefore been incorporated to support KG-enhanced clinical prediction by linking heterogeneous EHR codes to shared medical concepts via structured relations. However, existing KG-enhanced approaches remain limited in two aspects: (i) task-specific knowledge selection when extracting knowledge from a large multi-source MKG; and (ii) patient-level personalization and knowledge integration, where personalization is often weakly controlled and knowledge integration is not sufficiently aligned with longitudinal patient trajectories. To address these issues, we propose MedCPI, a unified Construct–Personalize–Integrate framework. MedCPI first performs task-guided schema induction and KG normalization to build a task-specific Concept MKG as a denoised knowledge pool, then constructs controlled patient-level PKGs via local expansion and short path search, and finally integrates PKG representations with time-aware EHR representations via cross-attention for prediction. Experiments on MIMIC-III and MIMIC-IV across four clinical prediction tasks show consistent improvements over strong EHR-only and KG-enhanced baselines. Ablations and additional analyses further validate the contribution of each stage and illustrate how MedCPI utilizes structured medical knowledge.