The other response assumes you're not really asking about "filing taxes" but about how to get the Corona virus stimulus payment.
I'll address "filing taxes", which will have a lot of "it depends".
The money you hear about people getting for just filing a tax return claiming kids requires them to have some earned income (wages or self employment). Without earned income, they are not eligible for the "refundable" Earned Income Credit or Additional Child Tax Credit. Both credits are calculated on the amount of earned income you have. No earned income means no "refund". A small amount of earned income means a small refund. The child tax credit does not "kick in" unless you have at least $2500 of earned income. So, a stay at home mom is unlikely to have any reason to file a tax return, with or without kids.
So, the next question is who should be claiming the kids. A child can be the “qualifying child” dependent of any close relative in the household. "Significant other" indicates a current relationship, as opposed to divorced or separated parents. If you and the other parent live together, either one of you (but not both) may claim the child. You may decide between you which one will claim the child. Only if you can’t agree, do the IRS tie breaker rules apply, to see who has first choice.
If your significant other claimed you, as a dependent, on both the 2018 and 2019 tax returns, you cannot get any stimulus in 2020.
