Target coverage problems have received much attention in recent years. In a large monitoring environment where targets are distributed over an entire monitored region, deploying static sensors leads to high hardware costs because a high number of sensors may be required to achieve network connectivity. This paper considers the target-patrol issue where a set of mobile data mules (DMs) are dispatched to efficiently patrol the given targets under a predefined time constraint. The targets are assigned weights indicating their importance, where more important targets should be visited more frequently by the DMs. Accordingly, this paper proposes a time-constrained weighted targets patrolling (TCWTP) algorithm for locally constructing efficient patrol paths, thereby ensuring globally stable intervals between visits to all target points. A performance analysis revealed that the proposed TCWTP mechanism outperforms existing works in terms of the average interval between visits, quality of monitoring satisfaction ratio, and monitoring fairness ratio.