He was forced to ask for their help 'on bended knee'. [=he was forced to beg for their help; he was forced to ask for their help in a very humble way]
If you want to clean the floor properly, you have to 'get down on your (hands and) knees' [=you have to kneel down on the floor] and start scrubbing.
She 'dropped/fell/sank to her knees' [=she knelt down] and begged for forgiveness. = She 'got down on her knees' and begged for forgiveness.
(..) It's not surprising he wouldn't know what a state he put you in. Sir, he's telling the truth. Let's say for the moment this Blatch does exist. You think he'd just fall to his knees and cry, "Yes, I did it. I confess. By the way, add a life term to my sentence." With Tommy's testimony, (..)1994 The Shawshank Redemption