In this paper, we investigate the impact of correlated noise on fast radio burst (FRB) searching. We found that (1) the correlated noise significantly increases the false alarm probability; (2) the signal-to-noise ratios (S/N) of the false positives become higher; (3) the correlated noise also affects the pulse width distribution of false positives, and there will be more false positives with wider pulse width. We use 55-h observation for M82 galaxy carried out at Nanshan 26m radio telescope to demonstrate the application of the correlated noise modelling. The number of candidates and parameter distribution of the false positives can be reproduced with the modelling of correlated noise. We will also discuss a low S/N candidate detected in the observation, for which we demonstrate the method to evaluate the false alarm probability in the presence of correlated noise. Possible origins of the candidate are discussed, where two possible pictures, an M82-harboured giant pulse and a cosmological FRB, are both compatible with the observation.