We are living amidst a pandemic caused by a ravaging coronavirus and an accompanying pandemic of misinformation that has strained our economy and socio-political institutions. A key scientific goal is to examine mechanisms that lead to the widespread propagation of contagions, e.g., misinformation and pathogens, and identify risk factors that can trigger widespread outbreaks. A common phenomenon 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. This work investigates spreading processes that entail evolutionary adaptations on random graphs with tunable clustering and arbitrary degree distributions. We derive a mathematical framework that predicts the epidemic threshold and the probability of emergence as functions of the characteristics of the spreading object, the evolutionary pathways of the pathogen/misinformation, and the structure of the underlying network as given by the joint degree distribution of single-edges and triangles. We supplement our theoretical finding with numerical simulations and case studies, shedding light on how clustering can offer pathways for mutation, thereby altering the course of the epidemic.