The object of the English sentence is indeed 'mother's dishes', but the object of the Swahili sentence is vyombo vyake ('her dishes'). In addition, the Swahili has a second "object", the benefactive ('mama'), which is licensed by the use of the applicative form of the verb (-safishia) rather than the normal transitive form (-safisha). So GunthardHa does provide a correct literal translation, but as he says, it sounds a bit odd in English, and the translation used here is to be preferred.
I agree with GunthardHa. If "nitamsafishia mama" means "I will clean for mother", and "vyombo vyake" means "his/her dishes", then this sentence literally translates to "I will clean for mother her dishes". For the sentence "I will clean mother's dishes for her", I would expect to see "Nitamsafishia vyombo vya(ke) mama"