With the acceleration of globalization, more and more people are willing or required to learn second languages (L2). One of the major remaining challenges facing current mispronunciation and diagnosis (MDD) models for use in computer-assisted pronunciation training (CAPT) is to handle speech from L2 learners with a diverse set of accents. In this paper, we set out to mitigate the adverse effects of accent variety in building an L2 English MDD system with end-to-end (E2E) neural models. To this end, we first propose an effective modeling framework that infuses accent features into an E2E MDD model, thereby making the model more accent-aware. Going a step further, we design and present disparate accent-aware modules to perform accent-aware modulation of acoustic features in a finer-grained manner, so as to enhance the discriminating capability of the resulting MDD model. Extensive sets of experiments conducted on the L2-ARCTIC benchmark dataset show the merits of our MDD model, in comparison to some existing E2E-based strong baselines and the celebrated pronunciation scoring based method.