Regularly offer thanks for foreign cheese (4)
I believe the answer is:
feta
'foreign cheese' is the definition.
(Greek cheese)
'regularly offer thanks' is the wordplay.
'regularly' indicates alternate letters (regularly take one letter, leave next etc.).
'thanks' becomes 'ta' (UK informal version of 'thank you').
The alternating letters of 'offer' are 'fe'.
'fe'+'ta'='FETA'
'for' acts as a link.
(Other definitions for feta that I've seen before include "Type of cheese" , "food there" , "Crumbly Greek cheese" , "Greek 8 across" , "White Greek cheese" .)