We conduct the convergence analysis of parameter estimation in the contaminated mixture of experts. This model is motivated from the prompt learning problem where ones utilize prompts, which can be formulated as experts, to fine-tune a large-scale pre-trained model for learning downstream tasks. There are two fundamental challenges emerging from the analysis: (i) the proportion in the mixture of the pre-trained model and the prompt may converge to zero during the training, leading to the prompt vanishing issue; (ii) the algebraic interaction among parameters of the pre-trained model and the prompt can occur via some partial differential equations and decelerate the prompt learning. In response, we introduce a distinguishability condition to control the previous parameter interaction. Additionally, we also investigate various types of expert structure to understand their effects on the convergence behavior of parameter estimation. In each scenario, we provide comprehensive convergence rates of parameter estimation along with the corresponding minimax lower bounds. Finally, we run several numerical experiments to empirically justify our theoretical findings.