With recent advances in deep learning, numerous algorithms have been developed to enhance video quality, reduce visual artefacts and improve perceptual quality. However, little research has been reported on the quality assessment of enhanced content - the evaluation of enhancement methods is often based on quality metrics that were designed for compression applications. In this paper, we propose a novel blind deep video quality assessment (VQA) method specifically for enhanced video content. It employs a new Recurrent Memory Transformer (RMT) based network architecture to obtain video quality representations, which is optimised through a novel content-quality-aware contrastive learning strategy based on a new database containing 13K training patches with enhanced content. The extracted quality representations are then combined through linear regression to generate video-level quality indices. The proposed method, RMT-BVQA, has been evaluated on the VDPVE (VQA Dataset for Perceptual Video Enhancement) database through a five-fold cross validation. The results show its superior correlation performance when compared to ten existing no-reference quality metrics.