Sent it letters showing how to authorise (7)
I believe the answer is:
entitle
'to authorise' is the definition.
(I know that entitle is a more specific form of the action authorise)
'sent it letters showing' is the wordplay.
'showing' indicates a hidden word (the answer is shown in the clue).
'ENTITLE' can be found hidden inside 'sent it letters'.
'how' is the link.
(Other definitions for entitle that I've seen before include "Give one the right - name" , "Name; empower" , "Style; give right to" , "Name, authorise" , "Warrant" .)