Jenny Whiteside, born 1812, natural daughter of Betty Whiteside is most likely a red herring. There is nothing to suggest that she was in any way connected to any of the John Whitesides around at the time. Her mother, Betty Whiteside was probably a "Singlewoman", a spinster, and not the Betty who was married to a John Whiteside. Baptism entry simply records "Jenny Whiteside, natural daughter of Betty Whiteside, abode Carleton". The term "natural daughter" means that she was illegitimate. Therefore she was not the daughter of John Whiteside & wife Betty, or of any other married couple. They could have belonged to any of the Whiteside families in the area or none. Vicars and Poor Law officials usually knew what was going on in their parish or made it their business to find out. If, as you appear to suggest, that Jenny was the daughter of Betty, wife of John, fathered by another man while Betty's husband was away doing his duty for King & country, that fact would not have gone unrecorded. I came across a similar case around the same time in anther parish near Poulton. The baptism entry read something like " Edward Smith, son of Ann Smith and Peter Brown. Notes: Ann Smith's husband is away serving with the 40th Foot." I researched illegitimate births of the era in that parish. Some single fathers were away in army or navy or worked as sailors or fishermen. Many married on their return. Some of course didn't survive. Others were just "passing through" and may never have come that way again, and may have been unaware that they'd fathered a child. The lists of baptisms I mentioned in a previous post, "Soldiers, Sailors & Strangers" is a snapshot of how things were at the time.
So my advice is to forget Jenny Whiteside, she just muddies the waters further. Jenny/Jane/Jennet was as common as Betty/Elizabeth.