I doubt that you can claim her even if her income was below the threshold because you are apparently breaking up and won't be living together for the whole year.
However, should you meet all the tests for claiming her, she can't claim her children. under IRC 152(b)(1) a taxpayer can not claim anyone as a dependent if they themselves can be claimed as a dependent.
as to the kids spending time with their father reg 1.152-1(b) says in part
The taxpayer and dependent will be considered as occupying the household for such entire taxable year notwithstanding temporary absences from the household due to special circumstances. A nonpermanent failure to occupy the common abode by reason of illness, education, business, vacation, military service, or a custody agreement under which the dependent is absent for less than six months in the taxable year of the taxpayer, shall be considered temporary absence due to special circumstances.
what this all means. if she earns over $4300 or leaves the household during the tax year you can not claim her. to her, her own children are qualifying children. if a person is a qualifying child that child can't be anyone else's qualifying relative. even if every test is met
by the way, the rules haven't changed over the last 5 years so if you claimed her children in the past. there could be issued if she were to go back and file amended returns for the open years
