  We reproduce the ${\cal N}=4$ supersymmetric mechanics on curved spaces,
constructed earlier within the Hamiltonian formalism, using the superfield
methods. We show that for any such mechanics, given by the metric and the third
order Codazzi tensor, it is possible to construct a suitable modification of
irreducibility conditions of linear ${\cal N}=4$ multiplets and obtain the
superfield Lagrangian by solving a simple differential equation. Also, we prove
that the constructed irreducibility conditions are consistent if and only if
the metric and Codazzi tensor satisfy the modification of the WDVV equations,
which are the conditions of existence of ${\cal N}=4$ supersymmetry.
