  We study the parameterized complexity of a variant of the $F$-free Editing
problem: Given a graph $G$ and a natural number $k$, is it possible to modify
at most $k$ edges in $G$ so that the resulting graph contains no induced
subgraph isomorphic to $F$? In our variant, the input additionally contains a
vertex-disjoint packing $\mathcal{H}$ of induced subgraphs of $G$, which
provides a lower bound $h(\mathcal{H})$ on the number of edge modifications
required to transform $G$ into an $F$-free graph. While earlier works used the
number $k$ as parameter or structural parameters of the input graph $G$, we
consider instead the parameter $\ell:=k-h(\mathcal{H})$, that is, the number of
edge modifications above the lower bound $h(\mathcal{H})$. We develop a
framework of generic data reduction rules to show fixed-parameter tractability
with respect to $\ell$ for $K_3$-Free Editing, Feedback Arc Set in Tournaments,
and Cluster Editing when the packing $\mathcal{H}$ contains subgraphs with
bounded solution size. For $K_3$-Free Editing, we also prove NP-hardness in
case of edge-disjoint packings of $K_3$s and $\ell=0$, while for $K_q$-Free
Editing and $q\ge 6$, NP-hardness for $\ell=0$ even holds for vertex-disjoint
packings of $K_q$s. In addition, we provide NP-hardness results for $F$-free
Vertex Deletion, were the aim is to delete a minimum number of vertices to make
the input graph $F$-free.
