Dodgy alibi, with connections offering gifts (9)
I believe the answer is:
abilities
'gifts' is the definition.
(as in talents)
'dodgy alibi with connections' is the wordplay.
'dodgy' indicates an anagram.
'with' is a charade indicator (letters next to each other).
'connections' becomes 'ties' (tie is a kind of connection).
'alibi' anagrammed gives 'abili'.
'abili'+'ties'='ABILITIES'
'offering' is the link.
(Other definitions for abilities that I've seen before include "The powers to do" , "skills" , "Talents" .)