The paper is devoted to the solution of a weighted nonlinear least squares problem for low-rank signal estimation, which is related to Hankel structured low-rank approximation problems. A modified weighted Gauss-Newton method (MGN), which uses projection on the image space of the signal, is proposed to solve this problem. The advantage of the proposed method is the possibility of its numerically stable and fast implementation. For a weight matrix, which corresponds to an autoregressive process of order $p$, the computational cost of iterations is $O(N r^2 + N p^2 + r N \log N)$, where $N$ is the time series length, $r$ is the rank of the approximating time series. Since the proposed algorithms can be naturally extended to the case of rank-deficient weight matrices, the method MGN can be applied to time series with missing data. For developing the method, some useful properties of the space of time series of rank $r$ are studied. The method is compared with state-of-the-art methods based on the variable projection approach in terms of numerical stability, accuracy and computational cost.