A high-density wireless sensor network can be deployed for specific information-gathering. In such a network, sensors need to route their sensed data to a base station, consuming highly-limited and unreplenishable energy resource. Hence, one of the most critical issues in designing sensor data gathering algorithms is to minimize the energy consumption for network longevity while meeting certain requirements given, such as delay constraints, which may vary depending on specific applications or environmental situations. We propose a novel two-phase clustering (TPC) scheme for energy-saving and delay-adaptive data gathering in wireless sensor networks. The proposed scheme partitions the network into clusters in phase I, each with a cluster head, forming a direct link between cluster member and cluster head. In phase II, each cluster member searches for a neighbor closer than the cluster head within the cluster to set up an energy-saving data relay link. The sensors use either the direct link or the data relay link for their sensed data forwarding depending on the requirements specified by the users or applications. Simulation results show that the TPC reduces average transmission distance by up to 75% with the help of data relay links when there is no delay constraint, implying that there is a significant amount of energy-savings in collecting sensed data.