Reconfigurable intelligent surface (RIS) assisted radio is considered as an enabling technology with great potential for the sixth-generation (6G) wireless communications standard. The achievable secrecy rate (ASR) is one of the most fundamental metrics to evaluate the capability of facilitating secure communication for RIS-assisted systems. However, the definition of ASR is based on Shannon's information theory, which generally requires long codewords and thus fails to quantify the secrecy of emerging delay-critical services. Motivated by this, in this paper we investigate the problem of maximizing the secrecy rate under a delay-limited quality-of-service (QoS) constraint, termed as the effective secrecy rate (ESR), for an RIS-assisted multiple-input single-output (MISO) wiretap channel subject to a transmit power constraint. We propose an iterative method to find a stationary solution to the formulated non-convex optimization problem using a block coordinate ascent method (BCAM), where both the beamforming vector at the transmitter as well as the phase shifts at the RIS are obtained in closed forms in each iteration. We also present a convergence proof, an efficient implementation, and the associated complexity analysis for the proposed method. Our numerical results demonstrate that the proposed optimization algorithm converges significantly faster that an existing solution. The simulation results also confirm that the secrecy rate performance of the system with stringent delay requirements reduce significantly compared to the system without any delay constraints, and that this reduction can be significantly mitigated by an appropriately placed large-size RIS.