On the one hand, denies spilling the oil (7)
I believe the answer is:
linseed
'oil' is the definition.
(I've seen this before)
'on the one hand denies spilling' is the wordplay.
'on' means one lot of letters go next to another.
'the one hand' becomes 'l' (abbreviation for left, as in the left-hand side).
'spilling' is an anagram indicator.
'denies' with letters rearranged gives 'inseed'.
'l' next to 'inseed' is 'LINSEED'.
'the' is the link.
(Other definitions for linseed that I've seen before include "Treatment for bats" , "Source of plant oil" , "Oil producer" , "Part of flax used as oil" , "Oil-producing plant" .)