I believe Francesca was a very young widow, afraid about the future as a single mother, and Cecil took advantage of the situation and persuaded her to let Jeff live with him so he would be raised as a Colby heir.
Her situation was based on insecurity and ignorance, and she thought she was doing the right thing.
Frankie... Acording to The Colbys story if memory serves:
1. Frankie told Jeff that she left the Colby family home while pregnant -- because she felt so ashamed of herself. She doesn't say what the circumstances were specific to that occassion. We don't know where she was going.(Jeff is the only one of Jason's children not born in the Colby family home; we don't know where he was born.) We don't even know if the household knew she was pregnant then. Nor do we know Cecil's whereabouts then; if he were still living at the Colby family home or not. At that time seemingly no one knew Jason & Frankie had slept together (but it happened when Jason was still single).
2. Twenty-eight years later, Adam came across a letter to Cecil from Philip in Cecil's Logan Rhinewood files. In it, Philip told Cecil that he knew that the child Frankie was carrying was not his. A handwriting expert later authenticated the letter as being written by Philip. Did Cecil get this letter before or after Frankie left the Colby family home, we don't know. Did Cecil ever know that it was Jason that Frankie slept with, we don't know. He may just have thought her a sluttish sister-in-law who was a fallen woman in his eyes and not fit to bring up a 'Colby' child -- given the mores of the time, he might have thought it better to let the world think Jeff was a Colby than let it be known that a Colby woman (even by marriage) was a harlot.
3. We do know that one of the things Jeff was mad about with Frankie was that she didn't bother to come get him from Cecil when she learnt that Philip had died. So, Frankie wasn't a widow when she'd first left Jeff with Cecil.
Frankie's response was that Cecil had promised that Jeff would get his share of the Colby fortune if she stayed out of their lives.
Connie says that Frankie didn't know that Philip had been disinherited -- and we know that happened before he joined the army. But Cecil did.
What if Cecil had confronted Frankie over Philip's letter, and told her Philip had been disinherited. Could he have wanted her out of their lives, but felt it wasn't fair to 'penalise' the innocent unborn baby and let the baby face an uncertain future, especially if Frankie hadn't much money of her own? And so he agreed to raise her unborn child as a Colby. What if Cecil got it out of her that it was Jason she'd slept with. What if for some peverse reason, it pleased him to raise Jason's child without Jason ever knowing he'd had a child by Frankie. So he 'bought' Frankie's silence in exchange for raising her child. ....Basically, there was enough left unsaid by various TC characters for the TC writers to fill in the blanks any way they wished in later seasons.
4. Frankie
was young. A mother before 20. She was 18 when Philip went off to war. She would have had no reason to fear the future as the Colby household, unaware of anything amiss, would have looked after her as a member of the family.
I think it was Philip's realisation that her unborn child wasn't his, and possibly her learning that Cecil knew too, which shook her and made her afraid of what the (social) repecussions would be. Her secret affair/ONS while a married woman wasn't a secret. The s*** would hit the fan sooner or later. Might have thought it was better to be thought of as a bad mother & wife than be in disgrace as a scarlet woman. And have Jason, Andrew's heir, possibly dragged in. ....A lot of it boils down to the mores of the time, and the social circles she was accustomed to being in. I agree Francesca thought she was doing the right thing.