Tenant not so keen without cover (6)
I believe the answer is:
lessee
'tenant' is the definition.
(I know that lessee is a type of tenant)
'not so keen without cover' is the wordplay.
'not so' becomes 'less' (eg 'not so big' means 'less big').
'without cover' says to take the centre (the letters which cover up the word are removed).
The middle letters of 'keen' are 'ee'.
'less'+'ee'='LESSEE'
(Other definitions for lessee that I've seen before include "One who rents" , "Leaseholder" , "I have to deal with letter" , "Hirer of property" , "Who pays the rent" .)