In a marine underwater object detecting system, sonobuoys collect underwater acoustic data and then wirelessly transmit it to an agile and controllable unmanned aerial vehicle (UAV), which, in turn, relays the raw data to the terrestrial computing center for further processing. However, constrained by the limitations in communication bandwidth and vehicle maneuverability, the UAV cannot fly over each sonobuoy to gather all the raw data within the prescribed period. In addition, to effectively detect the underwater object, we should also exploit diverse acoustical measurements of different sonobuoys (e.g., triangulation scheme) and maintain a certain level of throughput fairness among sonobuoys. Consequently, for efficient detection, both the bandwidth allocation and flight trajectory must be jointly optimized according to the importance of each sonobuoy. To this end, based on the α-fairness utility function, we design a bandwidth allocation strategy that assigns radio resource to a sonobuoy according to its weight. Similarly, in each searching round, the flying trajectory is also repeatedly optimized thus reducing the communication distances of those important sonobuoys. Moreover, in the proposed strategies, the weight of each sonobuoy and the α value are iteratively updated, which can progressively allow the UAV to fly more closely to those important sonobuoys to collect more effective data, thus significantly expediting the detecting process. Extensive simulation results have demonstrated the effectiveness of our proposed strategy, in that, compared with the benchmark strategies, it can detect underwater objects faster and more accurately.