Wrongly attempt to back current Conservative leader — it makes no sense (7)
I believe the answer is:
illogic
'it makes no sense' is the definition.
(lack of logic)
'wrongly attempt to back current conservative' is the wordplay.
'wrongly' becomes 'ill' (as in ill-advised, ill-gotten etc).
'attempt' becomes 'go' (go is a kind of attempt).
'to back' says the letters should be written in reverse.
'current' becomes 'i' (symbol for electric current).
'conservative' becomes 'c' (abbreviation commonly used in UK politics).
'go' written backwards gives 'og'.
'ill'+'og'+'i'+'c'='ILLOGIC'
'leader' acts as a link.
This may not be right. It may belong to another bit of the clue.
(Other definitions for illogic that I've seen before include "Senseless thinking" , "which makes no sense" , "false reasoning" .)