  We construct non-semisimple $2+1$-TQFTs yielding mapping class group
representations in Lyubashenko's spaces. In order to do this, we first
generalize Beliakova, Blanchet and Geer's logarithmic Hennings invariants based
on quantum $\mathfrak{sl}_2$ to the setting of finite-dimensional
non-degenerate unimodular ribbon Hopf algebras. The tools used for this
construction are a Hennings-augmented Reshetikhin-Turaev functor and modified
traces. When the Hopf algebra is factorizable, we further show that the
universal construction of Blanchet, Habegger, Masbaum and Vogel produces a
$2+1$-TQFT on a not completely rigid monoidal subcategory of cobordisms.
