Suppose, there's a legally allowed limit which a person seeking election can spend on his campaign. Money receivers and donors are disclosed, a legal campaign account has its limitations so a politician uses unaccounted money which can exceed legally reported expenditures by several times. In Russian, there's a phrase that can be literally translated as "a black cash register" (or, less literally, "black funds"). In English, can I say 'under-the-counter cash'? If not, how can I say otherwise?
What you're looking for is the expression off-the-books:
: not reported or recorded
// off-the-books transactions
// off-the-books covert operations
It applies to more than just money, but it certainly does apply there too.
For instance, you could say:
Their campaign was financed by off-the-books contributions.
In Indian English we use "Black Money" for money which is kept off the official books and on tax is intended to be evaded.
Specifically to your question, "Slush Funds" is a term that denotes a hidden or unaccounted sum of money that is kept aside for bribery or other illicit activities.
It has been used to refer to unaccounted funds used in election campaigns, as well as corporate funds reserved for bribes and lobbying where the source of funds is visible but destination is obscured.
[Choi] used her friendship with the South Korean leader to pressure businesses into donating to charitable foundations, which served as her personal slush fund.
"Slush Funds" has an interesting etymology, it is claimed to originate in the navy around 1830's from "the fat or grease (slush) skimmed from the top of the cauldron when boiling salted meat." The income thus earned from selling this unaccounted fat from a ship's galley would create an unofficial fund for small luxury expenses.
“Dark money” is the closest equivalent I can think of in English, American English in particular. However, in the United States, “dark money” is not necessarily illegal, it just implies that the source of the money is unknown (and would most likely want to remain unknown).
Not a perfect match, but I can’t think of anything closer to what you are looking for.
“Under-the-counter” tends to suggest more that you are avoiding taxes, and is not usually used to describe political donations.