I'd be amazed if "Jenny" didn't refer to generator. Note that the stories are set in the 1960, which may be relevant both to the slang used and the details of car construction.

Regarding "Special is looking into it, sir. I have the a p.p. as good as I could remember.", I'd presume that "Special" refers to Special Branch and would suspect that it was an abbreviation relevant to that context. The Metropolitan Police glossary tells me that APP can mean "application" or "Authorised Professional Practice" - would either of those fit?

BTW "As phony as a three-dollar bill." could be expressed in UK English, by those of a certain age and certainly in the 1960s, as "bent as a nine-bob note".