I know that he certainly had interactions with European powers in England's immediate vicinity such as France and the Holy Roman Empire, but did his foreign relations reach as far as what we now know as Russia?
No, he did not. England's relations with Moscow only truly began in 1554, when an English expedition to find a northern route to China through the Arctic failed, and ended with only one of the three original ships surviving, taking refuge at the mouth of the Dvina River, on Moscow's White Sea coast.
Source: The Merchants of Siberia: Trade in Early Modern Eurasia