Weakly supervised video anomaly detection (WSVAD) is a challenging task. Generating fine-grained pseudo-labels based on weak-label and then self-training a classifier is currently a promising solution. However, since the existing methods use only RGB visual modality and the utilization of category text information is neglected, thus limiting the generation of more accurate pseudo-labels and affecting the performance of self-training. Inspired by the manual labeling process based on the event description, in this paper, we propose a novel pseudo-label generation and self-training framework based on Text Prompt with Normality Guidance (TPWNG) for WSVAD. Our idea is to transfer the rich language-visual knowledge of the contrastive language-image pre-training (CLIP) model for aligning the video event description text and corresponding video frames to generate pseudo-labels. Specifically, We first fine-tune the CLIP for domain adaptation by designing two ranking losses and a distributional inconsistency loss. Further, we propose a learnable text prompt mechanism with the assist of a normality visual prompt to further improve the matching accuracy of video event description text and video frames. Then, we design a pseudo-label generation module based on the normality guidance to infer reliable frame-level pseudo-labels. Finally, we introduce a temporal context self-adaptive learning module to learn the temporal dependencies of different video events more flexibly and accurately. Extensive experiments show that our method achieves state-of-the-art performance on two benchmark datasets, UCF-Crime and XD-Viole