I am speaking, here, not in the more usual sense of a gap in the continuity of the core material. I mean, in reference to a rod of core material, that the outside diameter of the rod is sufficiently smaller than the inside diameter of the coil, that a uniform separation exists between the core and the coil.
The question is, does such a gap affect the inductance of such a system. I suppose a more general question would be how inductance would be affected if the coil were wound directly on a core whose permeability varies with diameter - that is, would such a system exhibit inductance different from a system of the same dimensions but with a core whose permeability is an average of the variable one.
To give a practical example, suppose I wind a coil on the outside of a piece of (nominal) 1/2 inch pvc pipe, whose outside diameter is 0.84 inches. I insert a rod of given permeability with an outside diameter of 0.622 inches (the inside diameter of nominal 1/2 inch pvc pipe).
Clearly, the inductance of such a system will be different than the case in which the coil is wound directly on such a core (smaller diameter coil, lower inductance), or in the case of a larger diameter rod (&c).
I have not found any material referencing such a system (possibly because the effects of such a system are so negligible as to be of no interest).
A further question might be of a system in which a layer of impermeable material fills the gap between the coil and a core of permeable material.