Video shot covering head of state's hanging (8)
I believe the answer is:
'hanging' is the definition.
(tapestry is a kind of hanging)
'video shot covering head of state's' is the wordplay.
'video' becomes 'tape' (I've seen this before).
'shot' becomes 'try' (I've seen this in other clues**).
'covering' is an insertion indicator.
'head of' suggests taking the first letters.
The first letter of 'state' is 's'.
'tapetry' placed around 's' is 'TAPESTRY'.
(Other definitions for tapestry that I've seen before include "Embroidered wall hanging" , "Woven wall-hanging" , "Complexity" , "Fabric picture" , "Woven hanging fabric with pictures" .)