Previous approaches for automatic lay summarisation are exclusively reliant on the source article that, given it is written for a technical audience (e.g., researchers), is unlikely to explicitly define all technical concepts or state all of the background information that is relevant for a lay audience. We address this issue by augmenting eLife, an existing biomedical lay summarisation dataset, with article-specific knowledge graphs, each containing detailed information on relevant biomedical concepts. Using both automatic and human evaluations, we systematically investigate the effectiveness of three different approaches for incorporating knowledge graphs within lay summarisation models, with each method targeting a distinct area of the encoder-decoder model architecture. Our results confirm that integrating graph-based domain knowledge can significantly benefit lay summarisation by substantially increasing the readability of generated text and improving the explanation of technical concepts.