It is a shitty app indeed, but in this case it was right to correct you. Despite what some people wrote, u/intricate_thing and u/Assassin21BEKA have a strong point. The variant you suggested would only be acceptable in colloquial spoken speech. It has to do with the functional sentence perspective. You see, even though Russian has a rather flexible word order, there are certain restrictions, and the most important ones come from the order you place your informational components in. In a neutral sentence, the things that stand at the end receive the logical stress. It's the rheme - the informational focus. Theme is everything that's less important and not highlighted. So, when you say 'Я выбежал из дома, взяв собаку", you highlight that the dog was with you when you ran out from your house, and should you change it to "Взяв собаку, я выбежал из дома", the focus would shift to the fact that you ran out.

In a neutral sentence (written one or spoken with a flat tone or with emphasis on the ending), the rheme would always be at the end. It is the general rule. There could be exceptions to it of course. For instance, you could use a rhematizer to move the rheme (the focus) to any part of the sentence. So, if you say "Только он сможет нам помочь", you stress the beginning with the rhematizer только, and the focus stays at the beginning. When speaking, you could also highlight various sentence parts using your intonation, giving them prominence even when they're not at the end. Different stresses make different implications. So, if you say "Это пожарная машина?" with a regular question intonation, you would be simply asking whether it is a fire department vehicle, but if you stress the word "пожарная", you would sound surprised that it's the firemen's car and not the cops' car, for instance. This works for statements and exclamations too, of course.

Another thing to consider is that in colloquial spoken speech it is rather common to use unnatural word order, speaking in somewhat inverted sentences, since people have less time to think and tend to give the most crucial information first, adding the less important details later (which can be negated by stressing the important bits with your intonation). E.g., in a text you would write "На главной площада стоит большая белая башня", whereas in a casual conversation you would quite often say "На главной площади стоит БАШНЯ (stressed), большая, белая". The main piece of information moves closer to the beginning, getting ahead of its attributes, but it's still highlighted with pitch, so the listener next to you would know what is the focus here and what is not.

Now, in your case, it is very unnatural to place the adverb of time at the end, since the main focus is the absence of object, not that it's absent AT THE MOMENT. It has some significance, but not enough to place it at the very end. The correct variant would absolutely be "У мамы его сейчас нет" or "У мамы сейчас его нет". In a spoken speech it would be okay-ish to say it like you did if you stress the word нет, otherwise it sounds very odd. If you continue the sentence, saying something like "У мамы его НЕТ сейчас, но на неделе ПОЯВИТСЯ", it would also be natural. Two-part sentence, where the first one informs the listener that the thing is not there, and the second one clarifies that it WILL be there this week.

I know it could be quite painful at first if your native tongue uses fixed word order and sets the focus just about where it wants to using the articles and other tools, but you should pick up the correct information placement naturally if you read enough texts written in good Russian. I would also recommend you to research the cons of duolingo and consider switching to a more comprehensive learning method, like the one described on Antimoon.com and Massimmersionapproach.com . These guides were written with regard to English and Japanese respectively, but you can use this approach to study just about any language. It is a much more fun and effective technique. Good luck with truly mastering the Russian language some day, if that's your goal!