Cable News Clips

Derived from Wiktionary,...

Wet your beak" is an idiom, often associated with gangster slang, meaning to take a share of the financial proceeds, profits, or benefits from a business deal or, frequently, illicit activity. It is used to describe getting a "cut" or a small portion of money, notably mentioned in mafia-themed media.
Wiktionary, the free dictionary

Usage Examples and Meaning:
Taking a Cut: It indicates a desire for personal profit. For example, "Before the deal closes, he wants to wet his beak".
Mafia/Gangster Context: Famous in mafia slang, meaning to collect a bribe or a share of protection money, as seen in The Godfather, where it refers to taking a small percentage of profits.
Less Common Literal Usage: Rarely, it can idiomatically mean to have a drink of a beverage.
Wiktionary, the free dictionary

Synonyms and Related Phrases:
Get a cut/slice of the pie.
Take a share.
Wet one's whistle (usually means to have a drink, though often confused).
Collect a kickback.
Merriam-Webster

The phrase implies a demand or expectation of profit for allowing a deal to proceed or for providing protection.
Wiktionary, the free dictionary