"Although Dmitri's unconventional ways and toleration of Catholics and Protestants bothered some, for almost a year he remained a popular tsar who pursued enlightened policies. He made, however two crucial errors. The first was neglecting to eliminate permanently the intrigues against him of the squat, bald, nearsighted Vasili Shuisky. The second was failing to deal effectively with the rising anti-foreign sentiments aroused by the behavior of many Poles who accompanied his future bride, the Polish Catholic Marina Mniszech, to Moscow in early May 1606." --Walter G. Moss, A History of Russia
So, let's say that the first Pseudo-Dmitri deals with these issues, and manages to become a mildly successful Tsar with a male heir. He was supported by Polish gentry, and so it would seem that he would gravitate toward friendship with Poland, while continuing Ivan IV's goals of expansion in Siberia and against Sweden. His reign would, if successful, avert the Dmitriads and the rise of the Romanov dynasty. What are the other effects of a Pseudo-Dmitriad Dynasty?