Baby at home, cooler close to that (6)
I believe the answer is:
infant
'baby' is the definition.
(I know that baby can be written as infant)
'home cooler close to that' is the wordplay.
'home' becomes 'in' ('he's home' can mean 'he's in').
'cooler' becomes 'fan' (I've seen this before).
'close to' says to take the final letters (the 'close' to the word).
The final letter of 'that' is 't'.
'in'+'fan'+'t'='INFANT'
'at' is the link.
(Other definitions for infant that I've seen before include "Rudimentary" , "Tiny baby" , "Child at an early stage of life" , "Young kid" , "Tot" .)