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I came across this sentence:
I can't understand why / by what rule it goes stuck after me?
As for me there should be a gerund here(not an infinitive without to):
Isn't with equal to while?
That is, although I did not find such a value , but can we say that it should be so:
Here are some examples, and then I'll try to deduce a rule from them.
Who'll look after you, with me stuck in the hospital like this?
is equivalent to:
Who'll look after you, as I am stuck in the hospital like this?
You could use the word "since" (in the sense of "because") instead of "as." You could also in this particular instance reword it with the word "while" - i.e. "... while I am stuck in the hospital like this...," but this doesn't always work. Consider:
He was a great football coach. With him having been fired for inappropriate comments, who will lead the team?
This is equivalent to:
He was a great football coach. As he has been fired for inappropriate comments, who will lead the team?
"Since he has been fired" would also work, but "while he has been fired" wouldn't make any sense here.
So, to attempt to describe the rule, I would say that "with + object pronoun + description of the state of the object" is equivalent to "as/since + subject pronoun + verb construction of the state described." But this is just my impression of how it works and my own attempt to formulate a rule.
Your first example is typical of informal speech:
Who will look after you, with me stuck in hospital?
You could change this to: ...with me lying in hospital but you CANNOT use stucking. There is no such word as stucking.
The verb is to stick, NOT to stuck.
And it makes no sense to speak of ..with me sticking in hospital.
The me third person object form follows the preposition with.
If you use the subordinating conjunction while, you require a different construction using the first person singular I.
Who will look after you while I am stuck in hospital.
