A game I've really gotten into for that sort of gritty, nuts-and-bolts mafia feel isn't a card game at all, but 'This Is the Police'. It's a resource management and decision-making video game that has this card-like quality to its story missions; you're shuffling your detectives around like a deck, handling gang rivalries, political pressure, and internal corruption. You get into the nitty-gritty of territory control, managing corrupt vs. loyal cops, and making deals with crime bosses that always come back to bite you. It's less about the romance of the mob and more about the grim, spreadsheet-and-murder logistics of it, which I find a lot of mafia fiction glosses over in favor of dramatic shootouts.
If we're strictly talking about card games with narratives, 'Griftlands' is a deck-building roguelike that has a whole faction dedicated to organized crime—the Admiralty's enforcers and the rival gangs you can play as or against. Your deck represents your reputation, your fighting style, and your negotiation tactics, and the story branches depending on who you ally with or betray. You really feel the instability of that life; one run you're a loyal soldier, the next you're playing both sides for your own benefit, and the game's systems do a great job of making those alliances feel consequential, not just cosmetic. The dynamics of trust and inevitable betrayal are the core of the underground crime loop it presents.
Neither of those are traditional 'card games' like you'd play at a table, but they use card mechanics to tell those stories in a way that feels deeper than most linear narratives. For a pure tabletop experience, the 'Android: Netrunner' universe, especially some of the older 'Android' board game material, dives into corporate espionage and syndicate warfare that has a very cyberpunk-mafia vibe, but it's more implied than the central focus.