Father over street going on to raise nipper (4,5)
I believe the answer is:
jack frost
'nipper' is the definition.
(Jack Frost is a personification of nipping cold)
'father over street going on to raise' is the wordplay.
'father' becomes 'fr' (abbreviation as in a priest's title).
'over street going' becomes 'ost' (I can't explain this - if you can you should believe this answer much more).
'on' says to put letters next to each other.
'to raise' becomes 'jack' (jacking is a kind of raising).
'fr'+'ost'='frost'
'frost' after 'jack' is 'JACK FROST'.
(Other definitions for jack frost that I've seen before include "Personification of freezing weather" , "Personification of winter weather that nips at your nose" , "Freezing weather personified" , "Cold weather, personified" , "The personification of cold weather" .)