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In this paper, we consider a relay-assisted wideband cognitive-radio (CR) system under the assumption that the frequency band chosen by the CR relay network for unlicensed spectrum usage overlaps with one or more bands dedicated to primary (e.g., licensed) narrowband links. Our objective is to optimize the performance of the CR system while limiting the interference in direction of the primary receivers, without requiring any adaptation of the transmitted signal spectra at the cognitive nodes. To this end, we study appropriate transmit power allocation (TPA) strategies among the cognitive relays. We first investigate the optimal centralized (OC) TPA solution and show that it can be formulated as a linear program. Since the OC-TPA solution requires a considerable amount of information exchange between the cognitive nodes, we develop two distributed TPA schemes, namely (i) a fully decentralized (FD) TPA scheme and (ii) a distributed feedback-assisted (DFA) TPA scheme. The FD-TPA scheme aims at maximizing the output signal-to-interference- plus-noise ratio (SINR) at the destination node of the CR network according to a best-effort strategy. It requires neither feedback information from the destination node nor an exchange of channel state information between the cognitive relays. The DFA-TPA scheme, on the other hand, utilizes feedback information from the destination node, in order to achieve a predefined target output SINR value, while minimizing the overall transmit power spent by the relays. Analytical and simulation-based performance results illustrate that notable performance improvements compared to non-cooperative transmission (i.e., without relay assistance) are achieved by the proposed schemes, especially when more than two hops are considered. In particular, the proposed distributed TPA schemes typically perform close to the OC-TPA solution.