With-construction: I can't work with all the children crying. "with + a participle construction" expresses a special circumstance of the main clause. Sometimes "with" can be dropped. But sometimes the connection between main clause and participle construction can become unclear as in this case:

I can't work, all the children crying.

Added: An old post of mine on another forum: with-construction (with + noun + participle)

If you don't know this special construction, you have difficulty to understand it.

A typical example: I can't work | with all the children crying. The sense of it is: I can't work when all the children are crying. This construction is very old and was already in use in Latin (ablativus absolutus).

You can choose whether you consider "crying" as a participle or gerund. Grammars treat it as participle. I tend to see it as gerund and explain the sentence as - I can't work with all the children (in the act of) crying.

Of course, it will take you quite a time till you can think in such constructions. The advantage of the with-construction is you can add in a shortened and elegant way circumstantial facts for which you normally need a subclause with a conjunction.

This is a whole chapter in grammars and you should have a lot of examples as there are some variants of this with-construction (My name as there is no standard term for this construction).