Mobile agentic AI is extending autonomous capabilities to resource-constrained platforms such as edge robots and unmanned aerial vehicles (UAVs), where strict size, weight, power, and cost (SWAP-C) constraints and intermittent wireless connectivity limit both on-device computation and cloud access. Existing approaches mostly optimize per-round communication efficiency, yet mobile agents must sustain competence across a stream of tasks. We propose a knowledge-driven reasoning framework that extracts reusable decision structures from past execution, synchronizes them over bandwidth-limited links, and injects them into on-device reasoning to reduce latency, energy, and error accumulation. A DIKW-inspired taxonomy distinguishes raw observations, episode-scoped traces, and persistent cross-task knowledge, and categorizes knowledge into retrieval, structured, procedural, and parametric representations, each with a distinct tradeoff between reasoning speedup and failure risk. A key finding is that knowledge exposure is non-monotonic: too little forces costly trial-and-error replanning, while too much introduces conflicting cues and errors. A UAV case study validates the framework, where a compact knowledge pack synchronized over intermittent backhaul enables a 3B-parameter onboard model to achieve perfect mission reliability with lower reasoning cost than both knowledge-free on-device reasoning and cloud-centric replanning.