Once a province reaches zero it becomes neutral. You would need to use governor (blue) cards to regain control through a vote.
This is wrong. The player's governor would not be removed and replaced with a neutral governor until the support check phase, which comes immediately after the card-playing action phase. If you had a yellow card to increase your support before moving on to the next phase, you could increase your support to 1 and retain control (provided there wasn't an active barbarian or enemy army in the capital, etc).
Refer to section 6.4 Support Check: "All of your Governors in provinces that now have a support level of zero or where the number of Mob counters equals or exceeds the support level are removed from office."