Assess offer a tender contains (4)
I believe the answer is:
rate
'assess' is the definition.
(I know that rate is a kind of assess)
'offer a tender contains' is the wordplay.
'contains' indicates a hidden word.
'RATE' is hidden in the letters of 'offer a tender'.
(Other definitions for rate that I've seen before include "Measure of frequency; merit" , "See 16" , "Adjudge" , "Bawl out" , "standard levy" .)