Great Western Coffee Shop

Plymouth to Chelmsford Essex, Route ^AP LONDN READING

^ I assume means its valid on TfL Underground between Paddington and Liverpool Street : Is Lancaster Gate a permitted route?

but why LONDN READING?
and what's AP?

AP = any permitted
Reading is listed because your route must go via Reading (SWT's Exeter - Waterloo route is not allowed, although SWT Reading-Waterloo would be,  likewise XC via Birmingham would be barred)
Londn means your route must also be via London.

Your question re Lancaster Gate is iperhaps nteresting.  It is a route that I have taken on a similar Bath - Stansted ticket without any problem (the ticket gate accepted my "gate pass" no problem - this was in the days when the NR tickets were the Eurostar size).  Now I have thought some more about it you are not allowed to break the Tube part of the journey on "+" tickets (you couldn't stop off at Oxford Circus for some shopping for example) so using Lancaster Gate is either technically against the rules and I was travelling illegally or there is a specific easement to allow it.  perhaps someone else knows?

The cross London part of a ticket is purely for that, crossing London to get the next station your train is going from.

So Walking from Paddington to Lancaster Gate and then getting the central line to Liverpool Street is illegal?

So I can go from any TfL station to any TfL station?  Or does it have to be Paddington - Liverpool Street, or can it be Lancaster Gate - Liverpool Street or even Waterloo - Tower Hill for example?

No, in this case it means Advance Purchase.

No.  You can make a single Tube journey from the London terminus you arrive in to the London terminus you depart from.

At least that is the general rule (which you should stick to until informed differently).  Hopefully someone will know if there are limited and reasonable exceptions to it when you are talking about Lancaster gate or Euston Square

AP in this case does not mean Any Permitted.  That is a common misperception - is actually stand for Advance Purchase.  You will never find AP on Anytime and Off Peak tickets.

Indeed, if you didn't pass through any barriers, you would be ok going down to Morden and back so long as you didn't meet any Revenue Staff.