carry on 后面何时加with何时不加with?

carry on是继续的意思，一楼是不是没搞明白？
carry on doing sth.[例]Carry on doing your homework.
carry on with sth.[例]Carry on with your homework.
用with的时候后面用名词

carry on sth 继续...
They managed to carry on their experiments in spite of the difficulties.
虽有困难,他们还是设法使试验继续下去。
carry on with sth 用。。。来继续。。。
They managed to carry on their experiments with their hard work in spite of the difficulties.
虽有困难,他们还是用自己的辛勤劳动设法使试验继续下去。
口语中的carry on指的是吵吵闹闹,举动时常;口语中的carry on(with sb)尤用于进行时态,指的是与某人有暧昧关系;一般的carry on的意思就是继续(做某事)，可以跟with sth/doing sth;或者是carry sth on

carry on sth 进行;继续
carry on with sth 用……来进行，继续

看具体语境的。
1，如果你要说明主语，或者要跟上一句的主语区别开，就要加上With，即携带的主人公。如：Please carry on that coat with you.
2，如果不必说明主语，主语就已经很明显，不会造成歧义。就不必加With。如Please carry on that coat.这样就成了祈使句了

表示继续/坚持做某事时用carry on with比如carry on with your work继续（做）你的工作表示保持某状态/某事物时用carry on如carry on the good job(保持事物）继续努力carry on doing sth(保持doing这个状态）继续做...

这要看后面有没有具体的携带者。
如：把这个苹果带上。 Carry on the apple.
你把这个苹果带上。 Carry on the apple with you.