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We study how massive ghost-free gravity $f(R)$-modified theories, MGFTs, canbe encoded into generic off-diagonal Einstein spaces. Using "auxiliary"connections completely defined by the metric fields and adapted to nonholonomicframes with associated nonlinear connection structure, we decouple andintegrate in certain general forms the field equations in MGFT. Imposingadditional nonholonomic constraints, we can generate Levi-Civita, LC,configurations and mimic MGFT effects via off-diagonal interactions ofeffective Einstein and/or Einstein-Cartan gravity with nonholonomically inducedtorsion. We show that imposing nonholonomic constraints it is possiblereproduce very specific models of massive $f(R)$ gravity studied in\cite{sarid,odintsr}. The cosmological evolution of ghost-free off-diagonalEinstein spaces is investigated. Certain compatibility of MGFT cosmology tosmall off--diagonal deformations of $\Lambda $CDM models is established.

Ghost-Free Massive $f(R)$ Theories Modelled as Effective Einstein Spaces and Cosmic Acceleration