Here is all your connections to mayflower passengers My Connections: Mayflower Passengers (wikitree.com)
check them out yourself
There are no Mayflower passengers descended from John Plantagenet [Plantagenet-143]. The closest connection you have to a Mayflower passenger (not counting the ones with "uncertain" lines) is 7th cousin, 10x removed. So they are distant cousins.
As for Mildred, she also does not descend from any Mayflower passengers. The closest connection is 7th cousin, once removed. So like you, she is a distant cousin to her Mayflower cousins.