Pet's first bit of supper - sponge (6)
I believe the answer is:
smooch
'pet's' is the definition.
(thesaurus)
'first bit of supper sponge' is the wordplay.
'first bit of' says to take the initial letters.
'sponge' becomes 'mooch' ('mooch' can be a synonym of 'sponge').
The initial letter of 'supper' is 's'.
's'+'mooch'='SMOOCH'
(Other definitions for smooch that I've seen before include "neck" , "Kiss and caress slangily" , "Canoodle" , "Snuggle" , "Kiss and cuddle (colloq.)" .)