We present a quasi-steady state reduction of a linear reaction-hyperbolic master equation describing the directed intermittent search for a hidden target by a motor-driven particle moving on a one-dimensional filament track. The particle is injected at one end of the track and randomly switches between stationary search phases and mobile nonsearch phases that are biased in the anterograde direction. There is a finite possibility that the particle fails to find the target due to an absorbing boundary at the other end of the track. Such a scenario is exemplified by the motor-driven transport of vesicular cargo to synaptic targets located on the axon or dendrites of a neuron. The reduced model is described by a scalar Fokker–Planck (FP) equation, which has an additional inhomogeneous decay term that takes into account absorption by the target. The FP equation is used to compute the probability of finding the hidden target (hitting probability) and the corresponding conditional mean first passage time (MFPT) in terms of the effective drift velocity V, diffusivity D, and target absorption rate λ of the random search. The quasi-steady state reduction determines V, D, and λ in terms of the various biophysical parameters of the underlying motor transport model. We first apply our analysis to a simple 3-state model and show that our quasi-steady state reduction yields results that are in excellent agreement with Monte Carlo simulations of the full system under physiologically reasonable conditions. We then consider a more complex multiple motor model of bidirectional transport, in which opposing motors compete in a “tug-of-war”, and use this to explore how ATP concentration might regulate the delivery of cargo to synaptic targets.