Chap gets dog, say, with hesitation (5)
I believe the answer is:
peter
'chap' is the definition.
(Peter is a man's name)
'dog say with hesitation' is the wordplay.
'dog say' becomes 'pet' (dog is an example).
'with hesitation' becomes 'er' (I've seen this in other clues).
'pet'+'er'='PETER'
'gets' is the link.
(Other definitions for peter that I've seen before include "Dwindle away to nothing" , "Boy's name (maybe blue)" , "till down under" , "prison cell" , "Disciple" .)