Near-threshold voltage (NTV) operation has the potential to improve the energy efficiency of digital integrated circuits. However, the use of a conservative timing guard band to avoid the timing errors introduces excessive timing margins, thus causing larger energy dissipation in the NTV region. An error-tolerant design based on timing error detection and correction circuits has been shown to be a promising solution to mitigate these issues. This paper presents a light-weight timing error-tolerant flip-flop (ETFF) design. This design detects timing errors using a node transition signal detector with only nine transistors and corrects these errors during the same clock cycle. Moreover, transistor sizing is explored to optimize the trade-off between performance and area overhead. The proposed ETFFs are inserted into a monitored circuit by replacing original flip-flops at timing-monitored points. To further reduce the overhead, we develop a mean-time-to-failure-aware method to select the monitored points by simultaneously considering the critical path coverage and activation rates of flip-flops. The simulation results show that a CNN accelerator using the proposed timing error-tolerant design implemented in the SMIC CMOS 40 nm process can robustly work at 1.1–0.3 V with only 3.5% area overhead. Furthermore, this design reduces the area overhead by 54.68% and improves the energy efficiency by 53.69% at 0.6 V, compared with the Razor flip-flop design. The advantage of the proposed design lies in that it requires smaller circuit overheads and can work reliably in a wider range of supply voltages.