In compressive sensing (CS) of images or videos, a block-based sensing or recovery scheme can facilitate low-cost sampling or recovery in memory and computation. However, its recovery with small block size and small subrate suffers greatly from its lack of information of the measurement data essential to recover a unique solution among many candidates. This study, based on prior knowledge of the signal to be sensed, namely, the relative magnitude difference of signal entries, designs a weighting process to limit the solution space of the recovered signal and combines it with much simplified Landweber iterations to deliver a complete recovery algorithm, called iterative weighted recovery (IWR). We theoretically verify the performance of the proposed IWR, including error bound, convergence rate, and stopping criterion. Application of the proposed IWR to block-based CS of images or videos confirms the quality improvement of the recovered images or videos and reduction of recovery time.