With the recent emergence of mixed precision hardware, there has been a renewed interest in its use for solving numerical linear algebra problems fast and accurately. The solution of least squares (LS) problems \(\min _x\Vert b-Ax\Vert _2\) , where \(A \in \mathbb {R}^{m\times n}\) , arise in numerous application areas. Overdetermined standard least squares problems can be solved by using mixed precision within the iterative refinement approach of Björck [BIT 7, 257–278 (1967)], which transforms the least squares problem into an \((m+n)\times (m+n)\) “augmented” system. It has recently been shown that mixed precision GMRES-based iterative refinement can also be used, in an approach termed GMRES-LSIR. In practice, we often encounter types of least squares problems beyond standard least squares, including the weighted least squares (WLS) problem \(\min _x\Vert D^{1/2}(b-Ax)\Vert _2\) , where \(D^{1/2}\) is a diagonal matrix of weights. In this paper, we present the FGMRES-WLSIR algorithm, a mixed precision approach for solving WLS problems, and discuss and analyze two different preconditioners.