Eager to figure in the publicity (4)
I believe the answer is:
avid
'eager' is the definition.
(synonyms)
'figure in the publicity' is the wordplay.
'figure' becomes 'vi' (I can't justify this - if you can you should give a lot more credence to this answer).
'in the' is an insertion indicator.
'publicity' becomes 'ad' (short for advert).
'vi' inserted within 'ad' is 'AVID'.
'to' is the link.
Can you help me to learn more?
(Other definitions for avid that I've seen before include "Keen, enthusiastic" , "'Eager, zealous (4)'" , "Keenly interested" , "deeply enthusiastic" , "passionate" .)