Estimate covers the first of the visitors (6)
I believe the answer is:
guests
'the visitors' is the definition.
(guest is a kind of visitor)
'estimate covers the first' is the wordplay.
'estimate' becomes 'guess' (guess is a kind of estimate).
'covers' indicates putting letters inside.
'first' suggests taking the first letters.
The initial letter of 'the' is 't'.
'guess' placed around 't' is 'GUESTS'.
'of' is the link.
(Other definitions for guests that I've seen before include "Company" , "They are associated with visiting" , "Visitors to whom hospitality is extended" , "House or hotel visitors" , "Folk receiving hospitality" .)