Here "hand" represents power or control. The phrase "seeking their lives" is an idiom that mean to want to kill someone. Alternate translation: "I will allow those who are wanting to kill them to defeat them" (See: [[rc://en/ta/man/translate/figs-metonymy]] and [[rc://en/ta/man/translate/figs-idiom]])
Here "hand" represents power or control. "Nebuchadnezzar ... his servants" names specifically "the ones seeking their lives." Alternate translation: "that is, I will allow Nebuchadnezzar king of Babylon and his servants to defeat the Egyptians" (See: [[rc://en/ta/man/translate/figs-metonymy]])
This can be stated in active form. Alternate translation: "Then after this, people will live in Egypt again" (See: [[rc://en/ta/man/translate/figs-activepassive]])