In terahertz (THz) massive multiple-input multiple-output (MIMO) systems, the combination of huge bandwidth and massive antennas results in severe beam split, thus making the conventional phase-shifter based hybrid precoding architecture ineffective. With the incorporation of true-time-delay (TTD) lines in the hardware implementation of the analog precoders, delay-phase precoding (DPP) emerges as a promising architecture to effectively overcome beam split. However, existing DPP approaches suffer from poor performance, high complexity, and weak robustness in practical THz channels. In this paper, we propose a novel DPP approach in wideband THz massive MIMO systems. First, the optimization problem is converted into a compressive sensing (CS) form, which can be solved by the extended spatially sparse precoding (SSP) algorithm. To compensate for beam split, frequency-dependent measurement matrices are introduced, which can be approximately realized by feasible phase and delay codebooks. Then, several efficient atom selection techniques are developed to further reduce the complexity of extended SSP. In simulation, the proposed DPP approach achieves superior performance, complexity, and robustness by using it alone or in combination with existing DPP approaches.