There are several reasons you can't claim the kids. They must live with you as a member of your household for the whole year. You can ignore temporary absences like a sleepover with friends, but you can't ignore the nights they live with their father, because they are also part of his household, so they don't live with you all year.
Also, because you aren't related, you can't claim the children if someone who is related who is also a taxpayer, could claim them. Their mother is a "taxpayer" for this purpose if her taxable income is more than $12,550, or if her income is less but she files a tax return to claim any other credit or benefit. And she can claim the children if they lived with her more than half the nights of the year. So if she is a taxpayer, you can't claim the children even if the lived in your home all 365 days of the year.
Your partner will likely be able to claim earned income credit by claiming the children as dependents, although she can't use head of household status.. and might not qualify for the full child tax credit (depending on her income).
