I think it's actually impossible to have the essence upgraded in your scenario, unless the girl's rarity is higher than common.So if I have a slave with Magical essence 4 and I want to get magical 5+, how much magical essence should client have. If he has like 1, it will get her child to 5? I don't know how it actually works
Quoting guyverek, the mod developer: "Upgrade algorytm: 7 + ([tier of species] * 8) - (same gender parent current essence level) * 6 + 5 * (opposite gender parent current essence level) percent".
So you have 7 + 1*8 (common rarity) - 4*6 (essence level of the girl) + 1*5 (essence level of the client) = -4%, which is the chance to have the upgrade. I don't know if a negative number increases the chance of downgrading the essence of the offspring (which happens when you lose the upgrade roll, and then "win" a 20% roll after that).
There also is a table in this comment that computes the chances for all possible essence pairings. The columns list the essence of the parent who has the opposite sex of the child you are calculating for, and in the rows you look for the other parent.
Hope I solved your questions!