  We show error estimates for a cut finite element approximation of a second
order elliptic problem with mixed boundary conditions. The error estimates are
of low regularity type where we consider the case when the exact solution $u
\in H^s$ with $s\in (1,3/2]$. For Nitsche type methods this case requires
special handling of the terms involving the normal flux of the exact solution
at the the boundary. For Dirichlet boundary conditions the estimates are
optimal, whereas in the case of mixed Dirichlet-Neumann boundary conditions
they are suboptimal by a logarithmic factor.
