Final answer:
Patriarch Filaret was the father of Mikhail Romanov, the first tsar of the Romanov dynasty. Filaret held considerable influence over state affairs during his son's reign as tsar.
Step-by-step explanation:
The relationship between Mikhail Romanov and Patriarch Filaret is that Filaret was his father. Mikhail Romanov was the first Tsar of the house of Romanov, becoming the ruler of Russia after the Time of Troubles. Filaret, his father, was formerly known as Feodor Nikitich Romanov and was forced to become a monk under the name Filaret, but later became Patriarch of Moscow and all Russia during his son's reign, exerting considerable influence over state affairs. The Romanovs came into power at a time following great instability and sought to reinforce the bond between the monarchy and the Russian Orthodox Church, enhancing the sacred authority of the tsar.