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<blockquote>When should a transformer core be potted?</blockquote>

Whenever you need some of<ul>

<li>Mechanical ruggedness.

<li>High dielectric strength.

<li>Minimize vibrations (and therefore audible whine) from capacitive, inductive, piezo-electric and magneto-restrictive effects.

</ul>

<blockquote>Can I place the high voltage conductors of the secondary coil on the inner layers considering the high dielectric strength of the PCB.</blockquote>

Since you are using two separate PCBs for the primary and secondary, this doesn't really accomplish anything.  You'll need at least one via, so there will be a point insulated only by the solder mask.  You could us a blind via, but that's more complicated, and cuts down on the layers you can use for the winding.

<blockquote>Air has a breakdown voltage of 3000 V/mm.</blockquote>

Not really.  There is no single figure for air breakdown voltage because there is no single spec for "air".  Variations in humidity make a large difference, and pressure also to a lesser extent.  1 kV/mm is a more reasonable figure to use in the absence of a particular spec or standard.  Creepage distance also matters since dirt and moisture can form a conductive path.

If this is something you plan to sell, then you need to look into the regulatory requirements and what kind of certification the customers or resellers will demand.  There are extra-stringent requirement for some applications, like medical.  Each standard will specify the minimum clearance and creepage distances for a particular voltage.

It's not clear whether both windings are low voltage with a high common mode difference, or if the secondary is a high voltage winding.  If the latter, you probably can't do this with PCB coils.  You won't be able get the turns ratio to go from something considered a "low" voltage to 10 kV.  You might be able to use a PCB coil for the primary, then many turns of fine wire for the secondary.  Either way, the power transfer will be "small".