Goddess having gravitas, on and off (4)
I believe the answer is:
gaia
'goddess' is the definition.
(thesaurus)
'gravitas on and off' is the wordplay.
'on and off' indicates alternate letters (letters selected on and off at regular intervals).
The alternate letters of 'gravitas' are 'GAIA'.
'having' acts as a link.
(Other definitions for gaia that I've seen before include "Earth personified" , "The earth viewed as a large organism (Lovelock)" , "Greek goddess of the Earth" .)