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.
(..) Smart people don't come here looking for the dead. So, you've met this Dragon queen, huh? And? She will only fight beside us if I bend the knee. You spent too much time with the Free Folk. Now you don't like kneeling. Mance Rayder was a great man. Proud man. The King Beyond the Wall and never bent the knee. How many of us people died for his pride? You still mad at us, boy? - You sold me to a witch. - A priestess. (..)2017 TvShow: Game of Thrones Title: Beyond the Wall Season: 7 Episode 6