This paper presents a novel narrowband interference (NBI) mitigation scheme for single carrier-frequency division multiple access systems. The proposed NBI cancellation scheme exploits the frequency-domain sparsity of the unknown signal and adopts a low complexity Bayesian sparse recovery procedure. At the transmitter, a few randomly chosen data locations are kept data free to sense the NBI signal at the receiver. Furthermore, it is noted that in practice, the sparsity of the NBI signal is destroyed by a grid mismatch between the NBI sources and the system under consideration. Toward this end, first, an accurate grid mismatch model is presented that is capable of assuming independent offsets for multiple NBI sources, and second, the sparsity of the unknown signal is restored prior to reconstruction using a sparsifying transform. To improve the spectral efficiency of the proposed scheme, a data-aided NBI recovery procedure is outlined that relies on adaptively selecting a subset of data-points and using them as additional measurements. Numerical results demonstrate the effectiveness of the proposed scheme for NBI mitigation.