The 6P-Cayley-Bacharach Point wrt a Circular Cubic (7P-s-Cu1) is the equivalent of the 8P-Cayley-Bacharach Point wrt a regular Cubic (9P-s-Cu1), because a Circular Cubic is a regular Cubic with two circular infinity points.
It can be compared with the relationship of a 5P-Pole wrt a Conic (5P-s-Tf5) and a 3P-Pole wrt a Circle (which also has two circular infinity points).
It can be compared with the relationship of a 5P-Pole wrt a Conic (5P-s-Tf5) and a 3P-Pole wrt a Circle (which also has two circular infinity points).
General background
A regular cubic is determined by 9 points. Through 8 random points infinitely many cubics can be drawn. However according to the Cayley-Bacharach theorem they all pass through a common point, called the Cayley-Bacharach point (8P-s-P1). Since a circular cubic is determined by 7 instead of 9 points, the Cayley-Bacharach Point for this type of Cubic is determined by 6 instead of 8 Points.
Construction
5P-s-Tf6(P) can be constructed using the regular Cayley-Bacharach construction of Hart with a small modification. If we replace the initial conics 12345, 12346, 12356 by circles 345, 346, 356, we get the CB-point of 3,4,5,6,7,8 and the two circular points. See Ref-64 and also Ref-34, QFG#3403.