Teacher is in possession of old and new assignments (8)
I believe the answer is:
missions
'assignments' is the definition.
(mission is a kind of assignment)
'teacher is in possession of old and new' is the wordplay.
'teacher' becomes 'miss' (I've seen this before).
'in possession of' means one lot of letters goes inside another (has inside).
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'and' means one lot of letters go next to another.
'new' becomes 'n' (common abbreviation eg NT for New Testament).
'o'+'n'='on'
'is' enclosing 'on' is 'ions'.
'miss'+'ions'='MISSIONS'
(Other definitions for missions that I've seen before include "Journeys for scientific, military or religious purpose" , "religious buildings - with 14a" , "Bombing raids" .)