ring-capacitor design
If the "connectors" are made of a conductive material then they become just part of each plate. They will not change the capacitance of the structure unless the thickness of the dielectric is different beneath the connector (hard to tell from your drawing).

As Dave Tweed said, the air gap serves no electrical purpose.

Of course, current flowing through the connectors is going to change the electric field between the plates. That's how capacitors work. The current flowing into one plate must be balanced by an equal current flowing out of the other plate. The relative voltage of the plate with current flowing into it will increase, with respect to the other plate.