It's not the best article you could find, but it was the first one which popped up in google which seemed to address describing the Cuban-American political clout in U.S. politics. (There's a ton of information available, all very strange!)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(snip)Morning in Miami
A New Era for Cuban Americans
By Alejandro Portes
Issue Date: 0.0.00
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T he recent visit of Pope John Paul II to Cuba gave the world an unusual glimpse of the last officially communist nation in the West. The implausible location of that country, just a short hop from American shores, was highlighted by the thousands of American pilgrims on the island for the occasion. The two old men who warmly greeted each other on the tarmac of José Martí International Airport could not be more different, but they also know each other. The pope knows communism intimately from his Polish and Eastern European days; Fidel knows the Church intimately from being brought up Catholic by Jesuit priests. They dueled with great finesse during those five days, trading subtle barbs at each other's ideology. Only on one point did they firmly coincide—the immorality of the United States's economic encirclement of Cuba and the need to bring it to an end. It was this common point that actually opened the doors of the island to the pope, creating a political turning point after almost four decades of iron dictatorship.
Only a few weeks earlier, the most prominent Cuban exile leader, Jorge Mas Canosa, had died of cancer in Miami. His death also marked a political turning point. It is but a slight exaggeration to say that under Mas's influence, Miami began to acquire the features of a Latin American dictatorship, opposite in ideology but similar in ruthless effectiveness to the communist regime that it opposed. The American trade embargo that the pope and Castro jointly denounced in Havana would probably have been lifted long ago had it not been for Mas and his powerful supporters. Although the pope overtly decried Washington's intransigence, his message was really aimed at Miami, where the roots of that intransigence lie.
While explicitly addressing the confrontation between two countries, the pope tacitly sought to reconcile the Cuban war across the Strait of Florida. For decades, charismatic leaders at both ends of the strait have used all available means to stay in power and crush their opponents. In Miami no less than in Cuba, the rule of law takes second place to the leaders' conviction in the rightness of their cause and their own historical role. Mas, like Castro, was a faithful practitioner of the old Latin American motto: For my friends, everything; for my enemies, the law.
It would be inaccurate, however, to portray Mas and his supporters as a criminal gang. He and the members of the organization that he created, the Cuban American National Foundation (CANF), are politically sophisticated extremists with a tight grip on the Cuban-American electorate. They direct Cuban Americans to vote overwhelmingly for hand-picked candidates, who purvey hard-line anti-Castro policies in Tallahassee and Washington. Farther afield, CANF's money has helped elect presidents and has heavily influenced the outcome of political struggles in Latin America and Africa, always on the conservative side. And, most importantly, the foundation has succeeded in imposing its views on the Clinton administration, literally dictating its policy toward Cuba. (snip/...)
http://216.239.57.104/search?q=cache:GULNm01UAPoJ:www.prospect.org/print/V9/38/portes-a.html+Miami+%22exiles%22+%2B+political+%2B+power+%2B+history&hl=en&ie=UTF-8(This is around 5 years old, but it should still represent the situation.)