The entire below from 1726 has, I think, a "P" for privilege postal marking. Before 1764 letters which went through the General Post free did not receive a special handstamp but were often cancelled with a manuscript "P" (rather than the postage charge). This "P" marking often does not look much like a 'P', as shown by illustrations in "Herewith My Frank" by JW Lovegrove.
The entire below is franked with "Frank E Harley Audt". Edward Harley had been an MP until 1722 (so would not get free postage in that regard), but was appointed joint Auditor of the Imprest for Life in 1702. The
Auditor of the Imprests was a profitable office of the Exchequer,
responsible for auditing the accounts of officers of the English crown
to whom money was issued for government expenditure. There is no definitive list of which offices had free franking privilege in 1726, but a list from 1838 does include the Audit Office, which replaced the Auditors of the Imprest when they were abolished in 1785.