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I have seen my father give some amount of money to my mother saying that it was for incidental expenses or asked her to spend on inessentials.

What is the right word for the money allotted/set aside for incidental expenses?

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    allowance – herisson Oct 16 '15 at 21:16
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    Are you looking for "pin money"? A small sum of money for spending on inessentials. (historical) An allowance to a woman from her husband for clothing and other personal expenses. oxforddictionaries.com/us/definition/american_english/pin-money – user66974 Oct 16 '15 at 21:18
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    In business contexts this is (or used to be) called petty cash. In domestic contexts, I can't think of a name for the money, but the container it's in (and, by synecdoche, the reserve itself) is call the kitty. – Dan Bron Oct 16 '15 at 21:27
  • Its container is canonically the cookie jar. (You might call such money cookie-jar money, but I don't think I've ever heard that.) – Drew Oct 17 '15 at 0:07
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    Contingency fund would be money set aside for emergencies or other unexpected expenses. But the term implies that the money should not be used for non-essentials. – Hot Licks Sep 20 '16 at 2:10
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spending money is an allowance for small personal expenses. It is extra money that you can spend on whatever you want, e.g. on ​activities you ​enjoy, ​entertainment, ​personal things, etc;

Example: How much ​spending ​money are you taking on ​holiday?

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Discretionary, as in discretionary spending, usually used to indicate non-essential

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    Could you offer some supporting evidence for your suggestion? It's always a good idea to include cited references, definitions or published examples, as appropriate, with your answers. – JHCL Oct 16 '15 at 23:17
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'Money set aside for incidentals' in a domestic context is sometimes called housekeeping, but this often includes groceries, so may not suit your purpose.

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Consider, pocket money.

: cash for day-to-day spending on incidental expenses - Princeton University, Farlex Inc

: (chiefly AmEng) A small sum of cash, carried on the person, for small, daily expenses Wiktionary

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A Nest Egg could be a candidate:

a fund of money accumulated as a reserve Merriam-Webster. "Paid for the computer out of his nest egg"

(idiomatic) A savings; a reserve of money. Wiktionary "Over time, they accumulated a tidy nest egg and retired comfortably."

A nest egg might imply a longer period of accumulation than allowance or emergency fund. But a quite useful expression.

Case Dough could be considered, too.

A small amount of money set aside for emergencies; mad money (1940s+)

[The Dictionary of American Slang, Fourth Edition by Barbara Ann]

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Petty Cash

In a business environment, the ubiquitous term for the pool of money, typically managed by a secretary or office manager, that one might spend on incidentals without having to fill an expense report, is Petty Cash.

I've also heard it used in non-business settings, but not as frequently referring to a pool of money, but more typically as an adjective describing the amount of money as being petty or trivial.

Example:

Yeah, Kobe is amazing. He bought that new whip with petty cash.

The OP's situation

That being said, in the example you provided, I believe petty cash would be a perfect fit. Imagine your Father saying:

Here Dear, put this with the petty cash.

or

Here Dear, add this to the petty cash.

Reference

From Merriam-Webster:

petty cash noun
Definition of petty cash
: cash kept on hand for payment of minor items

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