Off sb's hands: no longer the responsibility of sb 不再由某人负责

Now that the children are off my hands, I've got more time for other things.
现在我不用照顾孩子了,就有更多时间做其他事情.

On sb's hands

1.(指工作等)待处理,待办
I've got a work on my hands at the moment. 目前我手头有很多事要做.

2.由某人负责
I've got the neighbor's children on my hands this afternoon. 今天下午我得照看邻居的几个孩子.