Supervise six balls with date (7)
I believe the answer is:
oversee
'supervise' is the definition.
(I know that supervise can be written as oversee)
'six balls with date' is the wordplay.
'six balls' becomes 'over' (cricket).
'with' says to put letters next to each other.
'date' becomes 'see' (synonyms).
'over'+'see'='OVERSEE'
(Other definitions for oversee that I've seen before include "What superintendent will do" , "Be responsible for" , "Act as tutor" , "Watch people toil" , "risk eyestrain?" .)