A major hurdle in radio detection of cosmic-ray air showers is the continuous background that contaminates the signals. In this work, we use deep learning techniques to mitigate the effect of background by training two convolutional neural networks (CNNs). One network is used to distinguish the radio traces containing air-shower signals from those traces containing only background. The other network is trained to extract the underlying radio signals by removing the noise from the contaminated traces. In order to produce the required dataset for the training of the CNNs, we used CoREAS to simulate radio signals from cosmic-ray air showers for the geomagnetic field and observation height of the South Pole. As noise samples, we used radio background recorded by SKALA v2 antennas of a prototype station of the IceCube-Gen2 surface array at the South Pole. The frequency band used in the analysis ranges from 100\,MHz to 350\,MHz. The results show that the trained networks can indeed improve, on the one hand, the detection threshold of an externally triggered radio array and, on the other hand, the accuracy of the pulse parameters, such as the arrival time and amplitude of the radio pulses, which are subsequently used to reconstruct the properties of cosmic-ray air showers.