I have provided more than half of my children’s financial costs but may not necessarily have made more money than their father. How can I make sure their father does not get awarded the tax credi

[xmasbaby0]

Who do they live with?  Where did they spend at least 183 nights in 2019--with you or the other parent?

[Anonymous]

here are the important conditions that must be met - other than the ones apparently met 
1) be any of the following a) under age 19 at the end of the year b) a full-time student and under age 24 at end of the year c) or any age if totally and permanently disabled
2) have the same principal abode as the taxpayer for more than ½ the tax year  (183 nights). 

nothing can stop the father from claiming them on his return.   if both parents claim, then the one who files first can e-file, the one who files second will have the return rejected and must mail it.   if both file a return claiming the kids, then eventually both will get an IRS notice asking them to amend.  if neither does, the IRS will send a second notice asking each to prove proof.  the one with the better proof gets them and the other gets a bill from the IRS.