  Off-shell $(4,q)$ supermultiplets in 2-dimensions are constructed for
$q=1,2,4$. These are used to construct sigma models whose target spaces are
hyperk\"ahler with torsion. The off-shell supersymmetry implies the three
complex structures are simultaneously integrable and allows us to construct
actions using extended superspace and projective superspace, giving an explicit
construction of the target space geometries.
