This article presents the Zynq-embedded node for the Cherenkov telescope array (ZEN-CTA) node, a programmable system-on-chip (SoC) with White Rabbit (WR)-synchronization capability. It targets a solution for the uniform clock and trigger time-stamping module of the small-sized telescopes in the CTA. This module is tasked as a distributed acquisition device with a focus on obtaining time stamps for candidate Cherenkov events, which could be generated at potentially high rates from very-high-energy gamma rays and their subsequent distribution over Ethernet. In this context, the customized design of the ZEN-CTA node is examined thoroughly, including its generic implementation aspects and its main functional blocks. The design of the WR-assisted time-to-digital converters (TDCs) for time-stamping analog triggers is presented in detail alongside the implementation of an upgraded high-speed data path (1 Gb/s) for the WR-compatible Ethernet interfaces of the node. The new data path will feature a direct memory access engine for direct software transmissions and a hardware description language (HDL) coprocessor for high-speed forwarding. Next, the time-stamping accuracy of the WR-enhanced TDCs will be characterized alongside the forwarding efficiency of the new data path. Finally, conclusions are drawn, and the main contributions of this research are enumerated, a potential deployment within the CTA infrastructure to support the acquisition of Cherenkov light is considered, and additional use cases are mentioned.