In energy harvesting wireless sensor networks, ensuring complete targets coverage is a fundamental problem. In particular, targets are required to be monitored by at least one sensor node at all times. To date, past works have proposed to schedule sensor nodes alternately in the active/sleep state to maximize network lifetime whilst maintaining complete targets coverage, and affording sensor nodes recharging opportunities. However, they do not consider connectivity to the sink. We first propose a Linear Programming (LP) based solution to determine the activation time of sensor nodes. The design constraints include complete targets coverage, energy, and flow conservation to ensure data from sensor nodes monitoring targets are able to forward their data to the sink. We also propose an efficient heuristic algorithm as the LP solution requires an exhaustive collection of set covers. The heuristic iteratively selects sensor nodes to monitor targets and forward sensed data according to their residual energy. The simulation results show that the heuristic algorithm achieves 80% of the network lifetime computed by the LP solution at a fraction of the computation time.