In this research, we present the output-dependent anti-disturbance control of nonlinear systems with time-varying delay, where the system is impacted by three different multi-source disturbances. A hidden Markov model (HMM) is employed to identify asynchronous mode information between the system and the controller. To ensure that the system states approach the specified boundary in finite time, a dynamic output feedback controller and an adaptive anti-disturbance observer are designed. Also developed are sufficient conditions ensuring that internal type-2 (IT2) fuzzy Markov jump systems (FMJSs) achieve H∞ mean-square finite-time boundedness (H∞ MFTB). Based on this, the Finsler lemma is used to obtain the equivalent linear matrix inequalities (LMIs). Finally, two examples are used to demonstrate the practicability and effectiveness of the suggested strategy.