A modification of the harmonic superfield formalism in $D=4, N=2$ supergravity using a subsidiary condition of covariance under the background supersymmetry with a central charge ($B$-covariance) is considered. Conservation of analyticity together with the $B$-covariance leads to the appearance of linear gravitational superfields. Analytic prepotentials arise in a decomposition of the background linear superfields in terms of spinor coordinates and transform in a nonstandard way under the background supersymmetry. The linear gravitational superfields can be written via spinor derivatives of nonanalytic spinor prepotentials. The perturbative expansion of the extended supergravity action in terms of the $B$-covariant superfields and the corresponding version of the differential-geometric formalism are considered. We discuss the dual harmonic representation of the linearized extended supergravity, which corresponds to the dynamical condition of Grassmann analyticity.

Latex file, 17 pages with corrections of misprints