In an increasingly interconnected world, a key scientific challenge is to examine mechanisms that lead to the widespread propagation of contagions, such as misinformation and pathogens, and identify risk factors that can trigger large-scale outbreaks. Underlying both the spread of disease and misinformation epidemics is the evolution of the contagion as it propagates, leading to the emergence of different strains, e.g., through genetic mutations in pathogens and alterations in the information content. Recent studies have revealed that models that do not account for heterogeneity in transmission risks associated with different strains of the circulating contagion can lead to inaccurate predictions. However, existing results on multi-strain spreading assume that the network has a vanishingly small clustering coefficient, whereas clustering is widely known to be a fundamental property of real-world social networks. In this work, we investigate spreading processes that entail evolutionary adaptations on random graphs with tunable clustering and arbitrary degree distributions. We derive a mathematical framework to quantify the epidemic characteristics of a contagion that evolves as it spreads, with the structure of the underlying network as given via arbitrary {\em joint} degree distributions of single-edges and triangles. To the best of our knowledge, our work is the first to jointly analyze the impact of clustering and evolution on the emergence of epidemic outbreaks. We supplement our theoretical finding with numerical simulations and case studies, shedding light on the impact of clustering on contagion spread.