I honestly am not too sure. However, I can tell you that whether a recipient of money from a church has to include that as income on their personal income taxes is a super grey area. At a very minimum there is a total allowance of not including a “gift” of up to $14k per person. So you can do the math easily there... if there are two parents and three kids, you’re already up to $70k per year.
With the mobster question, I would think it is something like you contribute cash to a church or not for profit. The donor takes a charitable deduction there of 50% of the contribution. Then let’s say the church needs laundry services or meal services or building/contractor work done. They would engage a company owned by that mob affiliated person. Suddenly that money is cleaned.
The caveat to that is for not for profits that have to file 990s (any 501c3 other than a church or governmental agency), you have to disclose transactions with substantial contributors, board members, or other interested persons. Also, those contracts should be at fair market value. However, churches generally have less scrutiny and do not need to file that public disclosure form, so it could fly more under the radar. I’m not positive but that is what I would guess is the scheme for cleaning money.