In this paper, a quasi-average consensus problem is investigated in the cooperation-competition network, where the elements in the weight matrix of the network can be either positive or negative. To solve this problem, the whole network is firstly divided into two sub-networks, i.e., the cooperation subnetwork and the competition sub-network, and then a novel time-delayed event-triggered control scheme is designed in the competition sub-network. By establishing the solution of the multi-agent system, the ranges of the parameters of the event-triggered controller are determined, and the explicit expression of the error level is derived. Moreover, the convergence rate of quasi-average consensus is provided. It is found that Zeno behavior of the event-triggered controller can be excluded in our framework. Finally, simulation results are presented to validate the effectiveness of the theoretical analysis.