Assuming you are divorced or separated all of 2019, the custodial parent is the parent with whom the children lived for more than 182 nights of the tax year. The custodial parent is the one who can claim the children.

Yes, you can claim the children if they lived with you and not with the father and they meet all the requirements:

Qualifying child

• They're related to you.
• They aren't claimed as a dependent by someone else.
• They're a U.S. citizen, resident alien, national, or a Canadian or Mexican resident.
• They aren’t filing a joint return with their spouse.
• They're under the age of 19 (or 24 for full-time students).
  • No age limit for permanently and totally disabled children.
• They lived with you for more than half the year (exceptions apply).
• They didn't provide more than half of their own support for the year.

Note it doesn't say that you have to financially support them, therefore it doesn't matter that the father supported them. 

Please be aware that you can use your 2019 earned income to qualify for the Child Tax Credit. TurboTax will walk you through this.