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.
(..) a purposeful provocation by the Iranians to take advantage of the night's events and choke off the Western world's main supply of oil. Just as we're on our knees, they're gonna step on our throat. Aaron. I finally got past the gate. What's going on? I can't talk right now, Emily. (..)2016 TvShow: Designated Survivor Title: Pilot Season: 1 Episode 1