Tenant not so keen to give up housing (6)
I believe the answer is:
lessee
'tenant' is the definition.
(I know that tenant can be written as lessee)
'not so keen to give up housing' is the wordplay.
'not so' becomes 'less' (I've seen this before).
'to give up housing' says to take the centre (not the outside letters which house the rest).
The centre of 'keen' is 'ee'.
'less'+'ee'='LESSEE'
(Other definitions for lessee that I've seen before include "Who pays the rent" , "Leaseholder" , "Person renting a property" , "Hirer of property" , "Tenant who holds a lease" .)