You've got several mistakes in your sentence. First of all, the sentence is talking about one horse, not several. Genitive case goes after "нет", so "лошади" is "лошадь" in the genitive case in this sentence. Genitive case also follows "у", that is why it is "у нашей мамы. "Но" translates to 'but'. "Ну", which you wrote, is rather colloquial and translates to 'well'.
"But our mom has not a horse" Somebody can help me, why wrong is this? Thanks.