Oil a favourite thing played with hesitation (9)
I believe the answer is:
petroleum
'oil' is the definition.
(petroleum is a kind of oil)
'a favourite thing played with hesitation' is the wordplay.
I cannot quite see how this works, but
'favourite' could be 'pet' (synonyms) and 'pet' is found in the answer.
'hesitation' could be 'um' (sound made when hesitating) and 'um' is located in the answer.
The remaining letters 'role' is a valid word which might be clued in a way I don't understand.
This may be the basis of the clue (or it may be nonsense).
Can you help me to learn more?
(Other definitions for petroleum that I've seen before include "It helps in getting things moving" , "Fuel oil" , "Crude oil" , "Oil well product" , "Hydrocarbon oil" .)