Ultra-reliable and low-latency communication (URLLC) is one of the key use cases of the fifth generation (5G) wireless communications to facilitate specific application scenarios with stringent latency and reliability demands, such as industrial automation and Tactile Internet. A full-duplex (FD) relay with simultaneous transmission and reception in the same frequency band is an effective approach to enhance the reliability of cell-edge user terminals, by significantly suppressing self-interference (SI). However, the signal processing latency at FD relay due to SI cancellation, referred to as relaying latency, takes a significant part in the end-to-end latency, and therefore should be minimized, while guaranteeing high reliability. In this article, we first present an up-to-date overview of the end-to-end latency for an FD relay system, addressed on physical layer challenges. We investigate the possible solutions in the literature to achieve the goal of URLLC. The efficient solution is to allow a simple amplify-and-forward (AF) FD relay mode with low-complexity SI radio frequency and analog cancellations, and process the residual SI alongside the desired signal at base station in an adaptive manner, rather than being cancelled at relay in digital domain. Also, the residual SI can be utilized at base station to enhance the reliability and the degree of freedom in signal processing, not necessarily being cancelled as much as possible. The FD relay assisted system with adaptive SI utilization or cancellation enables extended network coverage, enhanced reliability and reduced latency, compared to the existing overview work.