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Adaptive detection of a range-spread target is addressed for a possibly singular estimated covariance matrix, in non-Gaussian clutter modelled as a spherically invariant random vector. Firstly, a modified generalised likelihood ratio test with recursive estimator (MGLRT-RE) is derived. To improve the adaptability and to reduce the computational complexity of MGLRT-RE, a simplified MGLRT (SMGLRT) is proposed and is proved to be constant false alarm rate (CFAR) to the statistics of the texture theoretically. Based on secondary data, the heuristic SMGLRT-CA (cell-averaging) and MGLRT-RE-CA are also designed. The SMGLRT outperforms the MGLRT and MGLRT-RE; similarly, the SMGLRT-CA with fully CFAR properties outperforms the MGLRT-CA and MGLRT-RE-CA. The performance assessment conducted by Monte Carlo simulation confirms the effectiveness of the proposed detectors.