Does any Marxist theory allow market economy the way the Chinese have it?
Did any Marxist author write anything on this happening in a socialist state?
Can a plan economy transform after a while, according to any Marxist theory?
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Without making this too long, the Chinese leadership still claims Marxist heritage (as you observed). Even for Deng Xiaoping's program there are mainland Chinese academics who found Marxist (or should I say merely pragmatic?) justifications for it:
The socialist market economy, as a part of Chinese Marxism, is both a heritage and a development of Marxist economics. In our prior understanding of Marxism, socialism is the opposite of capitalism. The basic nature of capitalism is private ownership, free market economic system, and wealth distribution according to the ownership of capital. As the opposite of capitalism, the basic nature of socialism lies in the public ownership of capital, planned economic system, and wealth distribution according to work. The former Soviet Union, some Eastern European countries, and China had tried for many years to follow these criteria for socialism, and the consequence is not good at all. This situation led the Chinese Communist Party to re-think and re-understand Marx and Engels, especially the ideas of their later years. If one inquires more deeply into why they contrasted socialism with capitalism, one will discover that in their understanding, the highest goal of socialism is to create the higher productive forces, to get rid of social inequality, to destroy poverty, and to make all social groups richer. Socialism is thus a more advanced system than capitalism. But these ideas are not easy to actualize. Each country has to find its own effective and possible way according to its own history and reality. Only when your socialist theory succeeds can it be proved to be true socialism, and only then can your practice be accepted and followed by your people. Otherwise socialism will have no reason and no power to attract the people. Here we should insist that practice is the only criterion to judge the truth of socialism and of Marxism.
On the other hand, critics point out that the Marxist-heritage claim looks rather silly for China nowadays:
China has come a long way from its revolutionary past since Deng Xiaoping’s free-market reforms put an end to Mao’s terrifying Leninist experiment in utopia. Today, it is all the more ridiculous to call an economy, the world’s second-largest, “socialist” when 70 per cent of it is privately owned, when it hosts the world’s largest army of billionaires, or when it grapples with issues such as a debt crisis, stock market woes and a real estate bubble.
What has been happening in China in the Reform era approximates in reality “state capitalism” more than “socialist market economy.” That is part of the reason there have been so much criticism and opposition from China’s socially conscious, progressive intellectuals, who do not reject the market but believe that the ideals of social justice from the revolution have largely been given up for the ideals of individual pursuit of self-enrichment.
Deng Xiaoping's reform was likened to Lenin's NEP (New Economic Policy) more than anything else, with the obvious downside of that, i.e. that it increased the number of capitalists, just as Lenin feared it would:
One of the greatest concerns among the Bolsheviks [regarding NEP] was “Who will win, the Capitalist or the Soviet Power?” (Lenin, 65). Indeed there was a validated fear of the possibility of the Capitalists taking over and reverting, once again, back to an Imperialist government and economy. As Lenin said in this particular document, “the capitalist, whom we are allowing to come in by the door, and even by several doors (and by many doors we are not aware of, and which open without us, and in spite of us),” (Lenin, 65). There were two options, according to Lenin: Either the capitalists take over and drive out the Communists or capitalism is utilized by the proletariats and the peasants, while submitting and serving the State (Lenin, 66). [...]
Although it is obvious, through deep analysis of his documents and the strong expression of his beliefs in Communism and against Imperialism and capitalism, it would seem to some that there was an almost paranoid anxiety of the capitalists. It’s not just that he acknowledges the dangers of capitalism and the inevitable chance that there will be those who will disagree with Communism, but there is undeniable evidence that this truly was a real fear for him. An example of this paranoia is when he is speaking about how the Soviet people will now have to work side by side with the capitalists, and how they will be hard to pick out of a crowd. But the fact that “They will squeeze profits out of you...” and that “they will enrich themselves, operating alongside of you” (Lenin, 72).