Sucker starts to lease another mine next to quarry (7)
I believe the answer is:
lamprey
'sucker' is the definition.
(aquatic animals with sucker mouths)
'starts to lease another mine next to quarry' is the wordplay.
'starts to' indicates taking the first letters.
'next to' is a charade indicator (letters next to each other).
'quarry' becomes 'prey' (synonyms).
The initial letters of 'lease another mine' is 'lam'.
'lam'+'prey'='LAMPREY'
(Other definitions for lamprey that I've seen before include "Long fish" , "seafood" , "Eel-like vertebrate with a sucker mouth and horny teeth" , "Fish-like vertebrate" , "sucker in the main" .)