Yellowface type II has two alleles. A budgie with only one YF II allele (single factor) will have a yellow face and a turquoise body colour after the first molt. A budgie with two YF II alleles (double factor) will have a yellow face, some yellow tail feathers and a blue body, which means double factor YF II's look like yellowface type I budgies. Sometimes the yellow spreads into the chest area a bit. But if the mother was a single factor YF II, and the father wasn't a yellowface of any kind then it's not possible to get a double factor YF II chick. Considering how patchy the yellow on the mother looked, I would guess that she's either a double factor YF II or a YF I. I would say YF I since a double factor YF II plus a non-yellowface budgie would give you 100% single factor YF II's (which look nothing like the mother or the chick).
The father is either a fullbody colour greywing or a poorly marked clearwing. Fullbody colour greywing is a co dominant mutation, (the father has one greywing allele and one clearwing allele) the only way to get fullbody colour greywing chicks is if the other parent is split for clearwing or greywing (if the father gave a clearwing allele, the mother would have to be split for greywing and if the father gave a greywing allele the mother would have to be split for clearwing since a budgie can't be split for fullbody greywing). There's only a 25% chance of getting a greywing. If the father is a clearwing and the mother is split for clearwing, there's a 50% chance of getting a clearwing chick. It's also possible to get a clearwing chick if the mother is split for clearwing and the father is a fullbody colour greywing. Breed the chick in question when she's old enough and you'll see what she is by what chicks you get from her and a greywing. If she's a clearwing, half the offspring will be clearwings and half will be fullbody greywings. If she's a fullbody greywing, half the offspring will be greywings and half will be fullbody greywings. If you get any greywings, you know she's not a clearwing.
Phewf! That was a lot of explaining. I hope I helped!