With the increasing installation of the high-voltage direct current device and flexible AC transmission system, the power transfer capability increases significantly. However, the risk of line overloads or even cascading failures caused by DC blocking or line outage also augments, which requires accurate and effective post-contingency corrective control (CC) actions. Traditional sensitivity-based methods are fast, but are less accurate and may suffer from the seesaw problem. Artificial intelligence optimisation-based methods usually exhibit better CC performance but are time-consuming. For fastly performing accurate CC actions, a second-order cone programming (SOCP)-based CC optimisation model for line overloads alleviation in meshed AC/DC power systems is proposed. Initially, to reduce the number of CC actions and optimisation variables (OVs), a heuristic search method is proposed to automatically select the most effective generators/loads for the overloaded lines. Subsequently, thyristor controlled series capacitor (TCSC) compensation levels and multi-terminal direct current (MTDC) power are treated as additional OVs to minimise load shedding/generation rescheduling. Additionally, to boost computational efficiency, SOCP format power flow equations for systems with MTDCs and TCSCs are proposed. Eventually, the effectiveness, accuracy, and superiority in computational efficiency of the proposed method are verified based on the modified IEEE 30-bus test system.