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To meet the practical requirements of low latency, low cost, and good privacy in online intelligent services, more and more deep learning models are offloaded from the cloud to mobile devices. To further deal with cross-device data heterogeneity, the offloaded models normally need to be fine-tuned with each individual user's local samples before being put into real-time inference. In this work, we focus on the fundamental click-through rate (CTR) prediction task in recommender systems and study how to effectively and efficiently perform on-device fine-tuning. We first identify the bottleneck issue that each individual user's local CTR (i.e., the ratio of positive samples in the local dataset for fine-tuning) tends to deviate from the global CTR (i.e., the ratio of positive samples in all the users' mixed datasets on the cloud for training out the initial model). We further demonstrate that such a CTR drift problem makes on-device fine-tuning even harmful to item ranking. We thus propose a novel label correction method, which requires each user only to change the labels of the local samples ahead of on-device fine-tuning and can well align the locally prior CTR with the global CTR. The offline evaluation results over three datasets and five CTR prediction models as well as the online A/B testing results in Mobile Taobao demonstrate the necessity of label correction in on-device fine-tuning and also reveal the improvement over cloud-based learning without fine-tuning.