Abdicate: To give up, to renounce
Abnegate: To deny, to refuse to concede
Sample 1: "The Minister abdicated his position owing moral responsibility for the incident"
Explanation: The minister gave up his position, renounced his position. You "Abdicate" what you have or what you are eligible for.
Sample 2: "The King Abdicated his throne."
Sample 1: "The counsellor abnegated the allegation of having taken kick-backs"
Explanation: The counsellor denied the allegation, refused to concede the allegation that he had taken kick-backs (bribes).
Sample 2: The teacher abnegated her responsibility for the child's failure.