On-board 4D cone-beam CT (CBCT) imaging using a linear accelerator (LINAC) is recently favored scanning protocol in image guided radiotherapy, but it raises the problems of excessive radiation dose. Alternatively, the 4D digital tomosynthesis (DTS) has been introduced for small-sized target localization, such as pancreas, prostate, or partial breast scan, which does not require a full 3D information. However, conventional filtered back-projection (FBP) reconstruction method produces severe aliasing artifacts due to sparsely sampled projections measured in each respiration phase within a limited angle range. This effect is even more severe when the LINAC gantry sweep speed is too fast to sufficiently cover the respiratory gating phase. Previous studies on total-variation (TV) minimization-based reconstruction framework would be an alternative approach to solve this problem, however, it presents the loss of sharpness during the iterations. In this study, we adopted an adaptively-weighted TV (AwTV) scheme which penalizes the images after the TV optimization. We introduced a look-up table which contains all possible weighting parameters during each iteration step. As a result, the proposed AwTV method provided better image quality compared to the conventional FBP and adaptive steepest descent-projection onto convex set (ASD-POCS) frameworks, showing higher structural similarities (SSIM) by factor of 1.12 compared to FBP and less root-mean-square error (RMSEs) by factor of 1.06 compared to ASD-POCS. The horizontal line profile of the spherical target inserted in the moving phantom showed that the images from FBP and ASD-POCS provided severe aliasing artifact and smoothed pixel intensities, but proposed AwTV scheme reduces the aliasing artifact while maintaining the object's sharpness. In conclusion, the proposed AwTV method has a potential for low-dose and faster 4D-DTS imaging, which indicates an alternative option to 4D-CBCT for small region target localization.