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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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]
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.
Yeah, Kobe is amazing. He bought that new whip with petty cash.
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.
Here Dear, add this to the petty cash.
petty cash noun
Definition of petty cash
: cash kept on hand for payment of minor items