I want to solve a least-squares problem of the form
$$\begin{array}{ll} \underset{x}{\text{minimize}} & \|Ax-b\|_2^2\\ \text{subject to} & \mathcal{L}(x)\succeq0\end{array}$$
with $\mathcal{L} : \mathbb{R}^n \to \mathbb{R}^{m \times m}$ being a linear operator.
This paper claims that they used Mosek to solve a problem of this form. To my best knowledge, the Mosek documentation on semidefinite programming does only include examples with linear objectives. How do I need to formulate the problem described above to solve it with Mosek?
You first write the problem as minimization of a new variable $t$ with the constraints $ \|Ax-b\|_2^2\leq t$ (and all your other constraints). The quadratic constraints can then be written using a second-order cone constraint as $\left|\left|\begin{matrix}1-t\\2(Ax-b)\end{matrix}\right|\right|\leq 1+t$. At that point, you have a mixed second-order and semidefinite cone program.
Having said that, in practice you would most often use a modelling language which would do this manipulations for you and then call Mosek.
You absolutely do not write the quadratic constraints as an LMI through a Schur complement as illustrated in the link in the comments. That's like trying to multiply two numbers by using logarithmic rules. Sure it's one way to do it if you have a calculator without any support for multiplication, but if your calculator has multiplication, use that button.