1
general::
Make a persistent effort regarding; also, persistently urge someone to do something. For example, We'll have to keep after the cobwebs, or He won't get anything done unless you keep after him. Also see KEEP AT, def. 2.
American Heritage Idioms
2
general::
Phrase(s): keep after someone (about something) [and] keep at someone (about something); keep on someone (about something); stay after someone (about something)
to remind or nag someone over and over to do something. • I’ll keep after you until you do it! • Mother stayed after Bill until he did the dishes. • She kept at him until he dried them and put them away. • We will have to keep on him about the report until he turns it in.
McGrawhill's American Idioms And Phrasal Verbs