Fuel that's a favourite outside (4)
I believe the answer is:
peat
'fuel' is the definition.
(peat is a kind of fuel)
'a favourite outside' is the wordplay.
'favourite' becomes 'pet' (synonyms).
'outside' indicates putting letters inside (some letters go outside others).
'a' placed into 'pet' is 'PEAT'.
'that's' is the link.
(Other definitions for peat that I've seen before include "Organic fuel" , "Turf as in bogland" , "Partially carbonised vegetable matter used as a fuel when dried" , "Saturated and compressed vegetable matter, used as fuel when dried out" , "Fuel from marsh" .)