We consider a dynamic network cascade process developed by Watts applied to a random networks with a specified amount of clustering, belonging to a class of random networks developed by Newman. We adapt existing tree-based methods to formulate an appropriate two-type branching process to describe the spread of a cascade started with a single active node, and obtain a fixed-point equation to implicitly express the extinction probability of such a cascade. In so doing, we also recover a special case of a formula of Hackett et al. giving conditions for certain extinction of the cascade.