Robots deployed in the real world over extended periods of time need to reason about unexpected failures, learn to predict them, and to proactively take actions to avoid future failures. Existing approaches for competence-aware planning are either model-based, requiring explicit enumeration of known failure sources, or purely statistical, using state- and location-specific failure statistics to infer competence. We instead propose a structured model-free approach to competence-aware planning by reasoning about plan execution failures due to errors in perception, without requiring a priori enumeration of failure sources or requiring location-specific failure statistics. We introduce competence-aware path planning via introspective perception (CPIP), a Bayesian framework to iteratively learn and exploit task-level competence in novel deployment environments. CPIP factorizes the competence-aware planning problem into two components. First, perception errors are learned in a model-free and location-agnostic setting via introspective perception prior to deployment in novel environments. Second, during actual deployments, the prediction of task-level failures is learned in a context-aware setting. Experiments in a simulation show that the proposed CPIP approach outperforms the frequentist baseline in multiple mobile robot tasks, and is further validated via real robot experiments in environments with perceptually challenging obstacles and terrain.