Give up on dog leads permanently (3,4)
I believe the answer is:
for good
'permanently' is the definition.
(synonyms)
'give up on dog leads' is the wordplay.
'give up' becomes 'forgo' (to forgo something is to give it up).
'leads' suggests taking the first letters.
The first letters of 'on dog' is 'od'.
'forgo'+'od'='FOR GOOD'
(Other definitions for for good that I've seen before include "Permanently, always" , "Purpose of charitable work" , "Informally, once and for all" , "on the side of the angels" , "Permanently - that's not bad" .)