Is there some difference of meaning about the remaining time in these sentences:
The airport is a long way from here, we have little time.
The airport is a long way from here, we haven't got much time.
Which way is more common?
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It's the exact same thing. The verb have in the present tense has two forms in English: have and have got=same thing.
So, we have little time = we haven't got much time.
The two forms of have mean exactly the same thing.
I have little time = I have got or I've got little time. I haven't got much time = I don't have much time.
And please don't believe all these false ideas about have got being British and have being American. It is simply not true.
Your question shows the declarative of have plus the negative of the same verb using the have got form.