Capacitance between primary and secondary sections is significant in Audio transformers because it is multiplied by the turns ratio of the sections concerned in a manner similar to the Miller effect in the triode valve.
The solution to this problem is the same: an earthed electostatic screen, usually made of foil, is interposed between the affected windings. We now have capacitance to earth, but the effect of this capacitance is minimal.
It is most important that the two ends do not contact electrically, as this would constitute a shorted turn. An electrostatic screen between primary and secondary is often fitted to mains transformers for rather different reasons.
If the insulation were to break down between primary and secondary, mains voltage would be connected to the secondary, which would be a safety hazard. By interposing an electrostatic screen , the fault current flows directly to earth and blows the mains fuse, thus making the equipment safe.
The electrostatic screen prevents RF inteference on the mains from being capacitivley coupled to the following circuitry.
In audio, the significance of RF intererence cannot be over emphasized, and this is sufficient incentive for using and electrostatic screen.
ref - Valve Anplifiers: Morgan Jones page 128
Note on "gapped" iron core inductors and transformers.
A gapped core will significantly increase inductance over a comparable air cored coil but because the air-gap is effectively a large magnetic resistance in the path of the magnetic flux it "swamps" the variation in permeability of the much smaller magnetic resistance of the core and inductance is more nearly constant.
Iron-cored inductors passing DC are invariably "gapped" in order to maintain their inductance up to their rated maximum current.
They will therefore have significant leakage flux which may intefere with nearby circuitry.
ref - Valve Anplifiers: Morgan Jones page 135