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This paper considers resource constrained path planning for a Dubins agent. Resource constraints are modeled as path integrals that exert a path-dependent load on the agent that must not exceed an upper bound. A backtracking mechanism is proposed for the Hybrid-A* graph search algorithm to determine the minimum time path in the presence of the path loading constraint. The new approach is built on the premise that inadmissibility of a node on the graph must depend on the loading accumulated along the path taken to arrive at its location. Conventional hybrid-A* does not account for this fact, causing it to become suboptimal or even infeasible in the presence of resource constraints. The new approach helps "reset'' the graph search by backing away from a node when the loading constraint is exceeded, and redirecting the search to explore alternate routes to arrive at the same location, while keeping the path load under its stipulated threshold. Backtracking Stopping criterion is based on relaxation of the path load along the search path. Case studies are presented and numerical comparisons are made with the Lagrange relaxation method to solving equivalent resource-constrained shortest path problems.