This paper studies the adaptive trajectory design of a rotary-wing UAV serving as a relay between ground nodes dispersed in a circular cell and a central base station. Assuming the ground nodes generate uplink data transmissions randomly according to a Poisson process, we seek to minimize the expected average communication delay to service the data transmission requests, subject to an average power constraint on the mobility of the UAV. The problem is cast as a semi-Markov decision process, and it is shown that the policy exhibits a two-scale structure, which can be efficiently optimized: in the outer decision, upon starting a communication phase, and given its current radius, the UAV selects a target end radius position so as to optimally balance a trade-off between average long-term communication delay and power consumption; in the inner decision, the UAV selects its trajectory between the start radius and the selected end radius, so as to greedily minimize the delay and energy consumption to serve the current request. Numerical evaluations show that, during waiting phases, the UAV circles at some optimal radius at the most energy efficient speed, until a new request is received. Lastly, the expected average communication delay and power consumption of the optimal policy is compared to that of static and mobile heuristic schemes, demonstrating a reduction in latency by over 50 and 20, respectively.