We have more time with after-school activities. (误) We have more time for after-school activities. (正) 这句话的含义是:有更多的课后活动时间。for 表示 “用途和目的”,介词词组作 time 的定语。
with 介词短语也可以用在名词之后作定语,但无法表示 “用途和目的”。一般表示:“涉及;与...有关的;具有” 之意。例如: a lot of trouble with him 与他有关的麻烦 a problem with money 涉及金钱的问题 a country with a long history 一个历史悠久的国家