Artist Paul with name on former partner's hanky (7)
I believe the answer is:
kleenex
'hanky' is the definition.
(I know that Kleenex is a type of hanky)
'artist paul with name on former partner's' is the wordplay.
'artist paul' becomes 'klee' (I've seen this in another clue).
'with' means one lot of letters go next to another.
'name' becomes 'n' (common abbreviation e.g. on forms).
'on' says to put letters next to each other.
'former partner' becomes 'ex' (an ex-partner).
'klee'+'n'+'ex'='KLEENEX'
(Another definition for kleenex that I've seen is " Disposable tissue brand".)