Hi, Tater,

"hanky" is a diminutive of "handkerchief".
A handkerchief is a small piece of cotton. linen, silk, or other
material -  often square - and it is used for the purpose of
wiping one's nose, eyes, face, etc. - or for decorative purposes.
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Apparently the girl next door to you has parents who are either remiss
in informing/educating their daughter - or who are themselves
uninformed.

Perhaps you could purchase a box of inexpensive cotton hankies

and give them to your neighbor, [You must be registered and logged in to see this link.]thus informing her and
providing her with hankies for the future. [You must be registered and logged in to see this image.]
Or, to save time, expense - and your nerves -  [You must be registered and logged in to see this image.]  you could extend a box
of tissues  to her and offer her several. 
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A hanky isn't much used by many. Kleenex has taken over.
I don't know anyone who actually uses a hanky. lol

Really? You do know what one is, right?

Yes, I know what one is! lol I have seen them, but kleenex has taken over, for a lot of people.

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Yup, pretty-much.
I still carry a handkerchief to almost any/all occasions
requiring attire more formal than my Levis and a bag smaller
than the one I "always" carry.

My son and most men I know always carry a handkerchief.
They use Kleenex for nose-blowing, [You must be registered and logged in to see this image.] - but they
always-always-always have a handkerchief in a pocket.

My father always carried a hankie too, but it's one tradition I never carried on because I didn't see the point of keeping a used snot rag in my pocket. It didn't seem very sanitary.

But if use kleenex (I call them tissues because I don't always buy the brand name, just whatever's on sale for a reasonable price), and just keep the hankie for other uses, like wiping your brow or to let a damsel in distress wipe her tears so she doesn't dilute her beer, well, that might be different.

Hi, Tater!
Well... then my Dad was your kind of guy, I think.
Lots of 'damsels' around wherever he was.