How do I remove hacked money?

Actually, there is a way, and it is Overkill approved.

If you have HoxHud, which is a HUD that is Overkill approved, it automatically has a feature that will detect if a payout bonus is too high. If it does detect it, it will prompt you after a game ends, saying:

Uh-Oh, Cheated Mission Alert!

Looks like you were in a mission with a cheater, would you like to reset your spending to $xxx, and your offshore to $xxx? Yes/No

And you will have the option to automatically remove "hacked" money. Here's an example below:

Buying and selling the same weapon over and over will slowly eat down the amount of spending cash. If you are below Infamy Level V, going infamous will also completely reset your spending cash to 0. (going to Infamy level VI and above will not reset your spending cash anymore) Those are the only reliable means to remove cash.

Note that just having excess cash (due to hackers or extremely good runs on White XMAS or Cook Off) will NOT trigger a cheater flag, and you're very likely to accumulate similar amounts of cash if you work your way up from Infamy V to Infamy XXV. Spending Cash also becomes stupidly easy to come by after Infamy I and one of the current methods of quickly leveling up also generates a nice amount of Spending Cash anyways.

[edit: as of PC update 100 (the skill tree overhaul), skills no longer have a money cost associated with them, so it is no longer possible to use them to remove spending cash obtained from a session that had a hacker.]