A constrained moving horizon observer is developed for nonlinear discrete-time systems. The algorithm is proved to converge exponentially under a detectability assumption with the data being exciting at all times. However, in many practical estimation problems, such as combined state and parameter estimation, the data may not be exciting for every period of time. The algorithm therefore has regularisation mechanisms to ensure robustness and graceful degradation of performance in cases when the data are not exciting. This includes the use of a priori estimates in the moving horizon cost function, and the use of thresholded singular value decomposition to avoid ill-conditioned or ill-posed inversion of the associated nonlinear algebraic equations that define the moving horizon cost function. The latter regularisation relies on monitoring of the rank of an estimate of a Hessian-like matrix and conditions for uniform exponential convergence are given. The method is in particular useful with augmented state space models corresponding to mixed state and parameter estimation problems, or dynamics that are not asymptotically stable, as illustrated with two simulation examples.