Instructed old king and old socialist (7)
I believe the answer is:
Here is my best explanation:
'instructed' is the definition.
(tutoring is a kind of instructing)
'old king and old socialist' is the subsidiary indication.
'old king' becomes 'tut' (King Tut).
'and' says to put letters next to each other.
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'socialist' becomes 'red' (red is a kind of socialist).
(Other definitions for tutored that I've seen before include "Instructed", "Privately taught", "instructed", "Rude tot (anag.)", "Taught small groups", "Coached", "Trained", "Gave private lessons to", "Was coach" and "Disciplined".)