If you look carefully at the 1833 marriage it says Plymouth or Plimmer [not Primer]
There's a couple of potential previous marriages as Plimmer in Manchester
1803 to Alice Thorp
1814 to Sarah Butterworth
The second one was widow and widower, so possibly the same man. There's a burial for an Alice Plimmor, wife of John, at All Saints, Newton (near Manchester) in 1808
A possibility is that Caroline is the daughter of Sarah's first marriage and considered John to be her father, using his name (but not found evidence of that as yet)