The method of alternating projections for extracting low-rank signals is considered. The problem of decreasing the computational costs while keeping the estimation accuracy is analyzed. The proposed algorithm consists of alternating projections on the set of low-rank matrices and the set of Hankel matrices, where iterations of weighted projections with different weights are mixed. For algorithm justification, theory related to mixed alternating projections to linear subspaces is studied and the limit of mixed projections is obtained. The proposed approach is applied to the problem of Hankel low-rank approximation for constructing a modification of the Cadzow algorithm. Numerical examples compare the accuracy and computational cost of the proposed algorithm and Cadzow iterations.