Welcome to the forum. You can not create an all quadrilateral mapped mesh around the arbitrary cut out. What you can do is create a rectangular region around the cutout and use that as a transition to an arbitrary quadrilateral mesh. I used an ellipse instead of you arbitrary shape, but the idea is the same.
For example,
reset
create curve location -5 0 0 location 5 0 0
curve 1 copy scale .75
curve 2 move y 5
create surface skin curve 1 2
create surface ellipse major radius 1 minor radius .5 zplane
body 2 move x 2 y 3
create brick bounding box Surface 2 extended percentage 50
imprint body all
del body 2 3
surface all size .2
surface 4 scheme submap # this will ensure the regular grid in most of the surface
mesh surface all
Thanks for your response.
Your example was helpful in sorting the issue.
As my surfaces are more complex and uneven I have subdivided the surfaces into more blocks and meshed them separately using submap and pave schemes based on the irregularity of surfaces.