Fast Fourier transform (FFT) and inverse FFT (IFFT) are used in orthogonal frequency-division multiplexing (OFOM) systems for efficiently converting frequency domain signals that carry discrete information symbols from or to time domain sequences, respectively. This discreteness of the frequency-domain symbol set in context of FFT is of great interest in this paper. In such a paradigm, unfortunately, sparse data corruption in the time domain sequence leads to catastrophic signal degradation in FFT due to its natural noise spread effect, and little can be mitigated by using error control coding schemes (ECC). Potential causes of sparse errors include time-domain sample loss, impulsive noise, and circuit failure in integrated circuits. In this paper, we first characterize the impact of sparse severe errors in the context of FFT, as opposed to that of commonly considered additive white Gaussian noise (AWGN). Based on observation of distinct impact, a series of computationally efficient algorithms are proposed for detecting and mitigating such errors in lieu of ECC. Simulation results show that our proposed algorithms can effectively combat various sparse errors and dramatically improve the performance.