Robots operating in unstructured human environments inevitably encounter failures, especially in robot caregiving scenarios. While humans can often help robots recover, excessive or poorly targeted queries impose unnecessary cognitive and physical workload on the human partner. We present a human-in-the-loop failure-recovery framework for modular robotic policies, where a policy is composed of distinct modules such as perception, planning, and control, any of which may fail and often require different forms of human feedback. Our framework integrates calibrated estimates of module-level uncertainty with models of human intervention cost to decide which module to query and when to query the human. It separates these two decisions: a module selector identifies the module most likely responsible for failure, and a querying algorithm determines whether to solicit human input or act autonomously. We evaluate several module-selection strategies and querying algorithms in controlled synthetic experiments, revealing trade-offs between recovery efficiency, robustness to system and user variables, and user workload. Finally, we deploy the framework on a robot-assisted bite acquisition system and demonstrate, in studies involving individuals with both emulated and real mobility limitations, that it improves recovery success while reducing the workload imposed on users. Our results highlight how explicitly reasoning about both robot uncertainty and human effort can enable more efficient and user-centered failure recovery in collaborative robots. Supplementary materials and videos can be found at: emprise.cs.cornell.edu/modularhil.