"I do not see a woman here."
Translation:Nie widzę tu żadnej kobiety.
Frankly, I disagree with the idea of calling "Tutaj nie widzę kobiety" a good translation. Sure, it's grammatical and so on, but putting "tutaj" at the beginning (and also at the end) gives an unusual emphasis. It's like "HERE I do not see a woman (but I do see women almost everywhere else".
So the rules I take away from here are:
1) "Tutaj" at the beginning or end of a sentence is OK if special emphasis is intended;
2) Elsewhere, "tu" and "tutaj" are interchangeable (though "tutaj" is safer for me because it looks less like the French for "ty");
3) In some cases, even native Polish speakers disagree amongst themselves ;-)
No you don't – „żadnej kobiety” would most naturally be translated as 'any woman' in this context and is accepted simply because 'a woman' can(through not always should, only if you need to stress that is some indefinite woman) be translated like that – Polish doesn't have articles, so in lieu of things like a/an and so on, we make do with demonstratives, determiners and pronouns. ;)