Check regularly offered by Armenians (4)
I believe the answer is:
rein
'check' is the definition.
(both can mean to control or limit)
'regularly offered by armenians' is the wordplay.
'regularly offered by' indicates alternate letters (letters taken at regular intervals).
The alternate letters of 'armenians' are 'REIN'.
(Other definitions for rein that I've seen before include "Strap to control horse" , "Restraint; strap attached to horse's bit" , "Horse controller" , "Leather used for control in horse-riding" , "Controlling strap" .)