In this paper, a cyclic-prefixed single-carrier (CPSC) transmission scheme with phase shift keying (PSK) signaling is presented for broadband wireless communications systems empowered by a reconfigurable intelligent surface (RIS). In the proposed CPSC-RIS, the RIS is configured according to the transmitted PSK symbols such that different cyclically delayed versions of the incident signal are created by the RIS to achieve cyclic delay diversity. A practical and efficient channel estimator is developed for CPSC-RIS and the mean square error of the channel estimation is expressed in closed-form. We analyze the bit error rate (BER) performance of CPSC-RIS over frequency-selective Nakagami-m fading channels. An upper bound on the BER is derived by assuming maximum-likelihood detection. Our simulation results in terms of BER corroborate the performance analysis and the superiority of CPSC-RIS over the conventional CPSC without an RIS and orthogonal frequency division multiplexing with an RIS.