"Locally Nonplanar" graph A 2-connected $3$-connected graph $G$ is "Almost Planar" Locally Nonplanar if it has a a $2$-connected spanning subgraph $H$ and an embedding in the plane such that $H$ is planar in this embedding and all the crossing edges are cords of faces of $H$ (all crossings and crossing edges are contained inside the faces of $H$.)
Is there any literature on this class of graphs? Are they classified by any other name? What kind of graphs are in this class? Are there graphs which are not Almost Planar?
Any information on this will be appreciated.
 A: 
OP: "Is there any literature on this class of graphs?"

The term "almost-planar graph" is already firmly in the literature:

Guoli Ding, Joshua Fallon, Emily Marshall.
  "On almost-planar graphs."
  arXiv abstract. Mar. 2016.

"A nonplanar graph $G$ is called almost-planar if for every edge $e$ of $G$, at least one of $G \setminus e$ and $G\,/\,e$ is planar."
"A graph $G$ is almost-planar if and only if $G$ is not $\{K_5, K_{3,3}\}$-free but for every edge $e$ of $G$, at least one
of $G \setminus e$ and $G\,/\,e$ is $\{K_5, K_{3,3}\}$-free."

          


          

Thm 1.1: The characterization of Gubser.



Gubser, Bradley S. "A characterization of almost-planar graphs." Combinatorics, Probability and Computing 5, no. 3 (1996): 227-245.

"We characterize the almost-planar graphs, those non-planar graphs for which 
$G \setminus e$ or $G\,/\,e$ is planar,
for all edges $e$ of $G$."
A: According to Kuratowski's theorem a planar graph is characterized by the absence of (subdivisions of) $K_5$ and $K_{3,3}$, where $K_5$ is the infamous pentagram and $K_{3,3}$ can be visualized as a hexagon with opposite vertices connected by edges and also resembles the smallest Möbius Ladder Graph.  
My suggestion to construct extremal graph "almost planar" graphs would be to fill the faces of a Fullerene Graph and augment its pentagonal faces to $K_5$ and the hexagonal faces to $K_{3,3}$. I don't think that those graphs have been described already and thus most likely have no name attached yet; maybe "Kuratowski Fullerene" would be acceptable.

The illustration shows how to fill the faces of Fullerene Graphs to construct almost planar graphs.
