A woman with her children was here ?????

A woman and her children were here.   用 were 是对的，但A woman with her children was here 是用was 而不是were。

许多人往往把with 当着 （和）的意思，结果把人数变成复数用了were来用。

With 有着《带着》的意思，重点是主角不是其他的人或事物。

A woman with her children was here 可翻译为一位妇人带着孩子在这里；不是一位妇人和她的孩子在这里。

看看以下的例子。

A car with three wheels was common in the olden days.

A table with four golden legs was stolen.

A house with three bed-rooms is well accepted by the people.