The baptism record for Hannah indicates mother is living in New Mill at the time. Cartworth is not generally contiguous so I only checked the census indexes for Fulstone, Wooldale, honley, Hepworth and Thurstonland.
Also, the township arrangments are a bit mangled in that area. When one township ceased to exist after the Black Death the land was divied up between Cartworth, Wooldale, hepworth and Fulstone so there are loads of (parts of) littered all over that side of the valley.
I can certainly try to get a copy of the baptismal register, although I cannot promise it will be legible; I struggled to read it. I already have the ones for Sam and John.
I can also double check the census records - I will pick up the reference from the index and check the fiche. Might take a couple of days, although I might also ge tover at lunch-time.
Do you know about the Huddersfield and Distrist Family History Society by the way?