The problem of content delivery in caching networks is investigated for scenarios where multiple users request identical files. Redundant user demands are likely when the file popularity distribution is highly non-uniform or the user demands are positively correlated. An adaptive method is proposed for the delivery of redundant demands in caching networks. Based on the redundancy pattern in the current demand vector, the proposed method decides between the transmission of uncoded messages or the coded messages of [1] for delivery. Moreover, a lower bound on the delivery rate of redundant requests is derived based on a cutset bound argument. The performance of the adaptive method is investigated through numerical examples of the delivery rate of several specific demand vectors as well as the average delivery rate of a caching network with correlated requests. The adaptive method is shown to considerably reduce the gap between the non-adaptive delivery rate and the lower bound. In some specific cases, using the adaptive method, this gap shrinks by almost 50% for the average rate.