Child at home with cooler temperature (6)
I believe the answer is:
infant
'child' is the definition.
(I know that infant is a type of child)
'home with cooler temperature' is the wordplay.
'home' becomes 'in' ('he's home' can mean 'he's in').
'with' means one lot of letters go next to another.
'cooler' becomes 'fan' (something which cools).
'temperature' becomes 't' (abbreviation).
'in'+'fan'+'t'='INFANT'
'at' acts as a link.
(Other definitions for infant that I've seen before include "Toddler" , "Small boy" , "Small baby" , "Tot" , "Very young child or baby" .)