It's been 3 years since you posted, so I don't know if you managed to solve it.
I have a William Pocock b 1817 to Charlotte Pocock in my tree but I'm not 100% sure if she is my Charlotte Pocock. Mine is born roughly around 1801 and I descend from a child from her marriage to William Crane.
If she is one and the same, then she was very young when she gave birth to William!
I would love to know what you have found out. I found out about the illegitimate child when I googled Kent parish records charlotte pocoock. Click on the first entry about petty sessions in Sevenoaks and scroll down to 1817 and hers is the first entry.