Low Earth orbit (LEO) augmentation in the global navigation satellite system has become a focus in the current satellite navigation field. To achieve high precision in positioning, navigation and timing services, relativistic effects should be considered, as they are difficult to distinguish from LEO satellite clock estimates and disturb their predictions. The relativistic effects on LEO satellite clocks are discussed in detail based on both theoretical and empirical results. Two LEO satellite clock prediction strategies are proposed, with and without removing the relativistic effect, using real data from typical LEO satellites: SENTINEL-3B and Gravity Recovery and Climate Experiment Follow-On (GRACE FO-1). For GRACE FO-1 and SENTINEL-3B, the relativistic effects are both on the order of nanoseconds and after removing the relativistic effects, the modified Allan deviations of the clocks are shown to be significantly improved. Based on the prediction strategies proposed, for SENTINEL-3B at around 810 km, with the prediction period increased from 30 to 3600 s, the root mean square error (RMSE) increases from 0.025 ns to about 1.4-1.6 ns. For the lower LEO satellite GRACE FO-1 at around 500 km, the RMSE of the predicted clocks increases more rapidly, i.e. from 0.012 ns at 30 s to about 4.5 ns at 3600 s. Results showed that the LEO satellite relativistic effects developed based on the theory could correct the majority, but not all of the once- and twice-per-revolution terms in the LEO satellite clocks. Although the corrections have exhibited effective improvements in the clock stability, they do not behave better than simply applying the mathematical model to the clock predictions. The latter model, however, does not have physical foundations as the former one.