  We construct the manifestly N=4 supersymmetric off-shell superfield "master"
action for any number n of the N=4 supermultiplets (4, 4, 0) described by
harmonic analytic superfields q^{+a}(\zeta, u), a= 1, ... 2n, subjected to the
most general harmonic constraints. The action consists of the sigma-model and
Wess-Zumino parts. We present the general expressions for the target space
metric, torsion and background gauge fields. The generic target space geometry
is shown to be weak HKT (hyper-K\"ahler with torsion), with the strong HKT and
HK ones as particular cases. The background gauge fields obey the self-duality
condition. Our formulation suggests that the weak HKT geometry is fully
specified by the two primary potentials: an unconstrained scalar potential
{\cal L}(q^+, q^-, u)|_{\theta = 0} which is the \theta = 0 projection of the
superfield sigma-model Lagrangian, and a charge 3 harmonic analytic potential
{\cal L}^{+ 3a}(q^+, u)|_{\theta = 0} coming from the harmonic constraint on
q^{+ a}. The reductions to the strong HKT and HK geometries amount to simple
restrictions on the underlying potentials. We also show, using the N=2
superfield approach, that the most general bosonic target geometry of the N=4,
d=1 sigma models, of which the weak HKT geometry is a particular case,
naturally comes out after adding the mirror (4, 4, 0) multiplets with different
transformation laws under N=4 supersymmetry and SO(4) R symmetry. Thus the
minimal dimension of the target spaces exhibiting such a "weakest" geometry is
8, which corresponds to a pair of the mutually mirror (4, 4, 0) multiplets.
