Vessel like that burning midnight oil, dash of paraffin thrown in (4,5)
I believe the answer is:
soup plate
'vessel' is the definition.
(dish for soup)
'like that burning midnight oil dash of paraffin thrown in' is the wordplay.
'like that' becomes 'so' (as in 'like so').
'burning midnight oil' becomes 'up late' (to burn the midnight oil is to stay up late).
'dash of' says to take the initial letters.
'thrown in' means one lot of letters goes inside another.
The first letter of 'paraffin' is 'p'.
'uplate' enclosing 'p' is 'upplate'.
'so'+'upplate'='SOUP PLATE'
(Other definitions for soup plate that I've seen before include "Item of crockery" , "Deep dish" , "Serving dish" .)