John F. Kennedy’s “Kennedy Doctrine” was really not anything terribly new. It did not change the relationship between the US and the USSR in any significant way. Nor did it change relations with Cuba. Relations with Cuba changed when Fidel Castro’s revolution overthrew the Batista government in 1959.
The Kennedy Doctrine really was just a restatement of the idea of containment, though it did so with special emphasis on Latin America. This did not change the relationship between the US and the Soviet Union. That relationship had long been tense. The US had been committed to the containment of communism at least since the announcement of the Truman Doctrine in 1947.
The Kennedy Doctrine also did not change the relationship between the US and Cuba. Cuba had had very close ties with the US under the regime of Fulgencio Batista. However, Batista was overthrown by Castro’s revolution at the beginning of 1959. Relations between the US and Cuba quickly soured after that and Cuba was closely allied to the Soviet Union before Kennedy entered office.
Thus, I would argue that the relationship between the US and the Soviet Union was essentially the same before and after the announcement of the Kennedy Doctrine. I would also argue that the Kennedy Doctrine did not fundamentally change the relationship between the US and Cuba.