Observations collected over time are often autocorrelated rather than independent, and sometimes include observations below or above detection limits (i.e. censored values reported as less or more than a level of detection) and/or missing data. Practitioners commonly disregard censored data cases or replace these observations with some function of the limit of detection, which often results in biased estimates. Moreover, parameter estimation can be greatly affected by the presence of influential observations in the data. In this paper we derive local influence diagnostic measures for censored regression models with autoregressive errors of order p (hereafter, AR(p)-CR models) on the basis of the Q-function under three useful perturbation schemes. In order to account for censoring in a likelihood-based estimation procedure for AR(p)-CR models, we used a stochastic approximation version of the expectation-maximisation algorithm. The accuracy of the local influence diagnostic measure in detecting influential observations is explored through the analysis of empirical studies. The proposed methods are illustrated using data, from a study of total phosphorus concentration, that contain left-censored observations. These methods are implemented in the R package ARCensReg.