Joanne,
No reference to Anne's burial, but the burials are only indexed for 1842 (when the register starts) to 1854, ie until civil registration of deaths began.
Clackmannan Parish included the villages of Sauchie, Fishcross, Kennet, Westfield and Forestmill, as well as Clackmannan itself, but I have not seen any reference to another churchyard/cemetery. I would suspect that Mary's abode at the time of her death might not be specified in the register because she was an illegitimate child - but that is no more than speculation.
David.