Is AC relay "inrush" current caused by more than the mutual inductance effect?
The &quot;plate&quot; is the relay armature.

Figure 1. Image source: D&amp;T Online.
When the armature is pulled in by the coil it closes the magnetic circuit. This results in an increase in coil inducatance and since the impedance (resistance to current flow) is given by \$ Z = 2\pi fL \$ it should be clear that for a given frequency the impedance will increase with the inductance. Your hunch is correct: the current will be high initially and reduce when the relay picks up.
This has the advantage that AC relays can be made to pull in strongly and automatically reduce their current when pulled-in. Be aware though that if the relay binds and remains energised while not pulling in that the coil may overheat and burn out. This can be a particular problem on larger contactors.

Is AC relay “inrush” current caused by more than the mutual inductance effect?

There's no &quot;mutual&quot; in a relay. Mutual inductance would require one coil affecting another - and there is no other in this case.


When the armature pulls in, it connects the two sides of the core - does this make the core in effect longer? That would explain the higher L!

No, it eliminates the air gap. Think of the air gap as a large magnetic circuit resistance due to air's poor permeability relative to the iron core. It's making a poor inductor. Close the air gap and now you have a good low reluctance magnetic circuit so the inductance increases.