Golfers that 'make the cut' usually get some amount of money, typically at least enough to cover their entry fees. The better you do, the more money you get. Some golfers get corporate sponsorships and endorsements.
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All the players, except for amateurs receive money. Only those who make the cut receive official money, those who miss the cut receive a small payment usually a few thousand, but it does not count towards the money list.
No, a term of accepting the sponsors invite is that he doesn't receive any official money, so no, he receives none of the tournament purse.
Those running the tournament pay the golfers, and they get their money from admission fees, tv fees and sponsor fees.