Usually, to be on hands and knees is to have both hands and knees on the ground; that is, to be on all fours in a crawling position. However, the expression to beg on hands and knees is meant as begging very strenuously and earnestly.
Strictly speaking, the "begging position" people in the West imagine is kneeling with hands clasped together; the person is not "on" the hands at all. The incorrect usage is tolerated because this is an idiom and people are familiar with the intent.
Example: Tom was begging on his hands and knees for Susan to come back to him, even though Tom had cheated on her with another woman. Susan's reply? "I'm calling my lawyer and getting a divorce!"