Toothless thank you note written by former PM (8)
I believe the answer is:
edentate
'toothless' is the definition.
(edentate means lacking teeth)
'thank you note written by former pm' is the wordplay.
'thank you' becomes 'ta' (informal way of saying thank you).
'note' becomes 'te' (musical note).
'written by' says to put letters next to each other.
'former pm' becomes 'Eden' (Anthony Eden, UK prime minister 1955-1957).
'ta'+'te'='tate'
'tate' after 'eden' is 'EDENTATE'.
(Other definitions for edentate that I've seen before include "anteater for one" , "Toothless mammal" , "Gummy" , "Sloth, perhaps" , "Animal as sloth, armadillo" .)