Single-photon LiDAR possesses weak echo sensing capability, but random backgrounds can easily cause target response loss and fragmented false alarms. In particular, sparse return photons, background photons, and dark counts may generate stochastic responses, making weak target preservation and false-alarm suppression difficult under single-photon observations. This paper proposes STMP-UNet, which uses a structure-aware target alignment module to suppress ineffective shallow response propagation through decoder semantic constraints, and combines a morphology prior module to support the spatial continuity of coarse target-region responses. The former selectively transfers target-related shallow information under high-level semantic guidance, while the latter introduces multi-scale structural constraints to alleviate spatial fragmentation of sparse target responses. Experiments are conducted on simulated single-photon data constructed from Anti-UAV infrared sequences. A photon-counting statistical simulation is used to reproduce sparse photon responses, and an infrared-derived proximity map provides auxiliary spatial structural information. Results show that compared to Pure U-Net, STMP-UNet improves Object F1 from 0.7115 to 0.7531 and Pd from 0.6635 to 0.7074, and reduces FPPI from 0.1907 to 0.1619. Ablation and candidate-level FROC analyses further demonstrate the effectiveness of the proposed structure modeling for preserving weak target responses and improving the detection–false-alarm trade-off.