  The problem of extending partial geometric graph representations such as
plane graphs has received considerable attention in recent years. In
particular, given a graph $G$, a connected subgraph $H$ of $G$ and a drawing
$\mathcal{H}$ of $H$, the extension problem asks whether $\mathcal{H}$ can be
extended into a drawing of $G$ while maintaining some desired property of the
drawing (e.g., planarity).
  In their breakthrough result, Angelini et al. [ACM TALG 2015] showed that the
extension problem is polynomial-time solvable when the aim is to preserve
planarity. Very recently we considered this problem for partial 1-planar
drawings [ICALP 2020], which are drawings in the plane that allow each edge to
have at most one crossing. The most important question identified and left open
in that work is whether the problem can be solved in polynomial time when $H$
can be obtained from $G$ by deleting a bounded number of vertices and edges. In
this work, we answer this question positively by providing a constructive
polynomial-time decision algorithm.
