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In next-generation communications, massive machine-type communications (mMTC) induce severe burden on base stations. To address such an issue, automatic modulation classification (AMC) can help to reduce signaling overhead by blindly recognizing the modulation types without handshaking. Thus, it plays an important role in future intelligent modems. The emerging deep learning (DL) technique stores intelligence in the network, resulting in superior performance over traditional approaches. However, conventional DL-based approaches suffer from heavy training overhead, memory overhead, and computational complexity, which severely hinder practical applications for resource-limited scenarios, such as Vehicle-to-Everything (V2X) applications. Furthermore, the overhead of online retraining under time-varying fading channels has not been studied in the prior arts. In this work, an accumulated polar feature-based DL with a channel compensation mechanism is proposed to cope with the aforementioned issues. Firstly, the simulation results show that learning features from the polar domain with historical data information can approach near-optimal performance while reducing training overhead by 99.8 times. Secondly, the proposed neural network-based channel estimator (NN-CE) can learn the channel response and compensate for the distorted channel with 13% improvement. Moreover, in applying this lightweight NN-CE in a time-varying fading channel, two efficient mechanisms of online retraining are proposed, which can reduce transmission overhead and retraining overhead by 90% and 76%, respectively. Finally, the performance of the proposed approach is evaluated and compared with prior arts on a public dataset to demonstrate its great efficiency and lightness.