This sentence has two distinct translations that make sense in English. Either "Mom doesn't have it [masculine noun] now", or "He is not at mom's place now".
In Russian, previously mentioned masculine nouns become "him", whereas in English they are "it". So we could have:
у неё ещё есть этот стол? Нет, у мамы его сейчас нет.
Does she still have that table? No, mom doesn't have it now.
However, у is also (informal?) for at that person's place (house/home). In Australian English we might say: "Where's your mum living now? At mine", where "at mine" means "at my house/home/place". So we could have:
Дима готовит для мамы? Нет, у мамы его сейчас нет.
Is Dima cooking for mom? No, he is not a mom's now.
"Mum doesn't have him" might be translated as «у мамы нет его», but Russian prefers to put pronouns before the verb. So it's more natural as «у мамы его нет».
Also, Russian prefers adverbs like "now" to also be before the verb (and after any pronouns), so we end up with «у мамы его сейчас нет»
In modern-day English, the negative forms of the verb "to have" cannot take a pronoun as a direct object without the support of "do/does." You can say, for example, "she hasn't a prayer now," or, "she hasn't the time now," but you cannot say, "she hasn't him now." It has to be, "she doesn't have him now."