There were at least two major challenges to communism in China during this decade.
First, there was the challenge posed by the decline and fall of communism in Europe. As communism fell apart in Eastern Europe, the Chinese communists had to chart some path that would allow their country to progress without the Party losing control as it had in the Soviet bloc. This was eventually accomplished by Deng Xiaoping, who managed to modernize the country and set it on its path to its current level of prosperity.
Second, there was the challenge posed by dissidents within the country. This is most clearly seen in the 1989 protests at Tiananmen Square. There were many in the country who were pushing for a more democratic system. This could have led to a loss of power by the communists just as it had in Europe.
In the 1980s, communism in China faced major challenges as the Cold War ended. The Party survived by promoting economic growth while still being quite repressive towards political dissidents.