If a 2019 return has not been filed and accepted, they will defer to the 2018 return for the stimulus payments. See the link below for additional information.
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If you are required to file a tax return (test #1 below) then he cannot claim your child(ren).
If you do not have enough income to require a tax return then he can claim them ONLY if he meets ALL of these requirements. (Test #2 & #4 in particular).
Even then, the only credit is the $500 "other dependent" credit.
---Tests to be a Qualifying Relative (& Unrelated Persons)---
(Must meet ALL of these tests to be a dependent)
1. The person cannot be your qualifying child or the qualifying child of any other taxpayer.
2. The person either must be related to you, or must live with you all year (all 365 days - There are exceptions for temporary absences such as school, illness, business, vacation, military service) as a member of your household.
3. The person's gross income for the year must be less than $4,200 (tax-exempt income, such as certain social security benefits, is not included in gross income)
4. You must provide more than half of the person's total support** for the year.
5. The person is not filing a joint return.
In any case, the person must be a U.S. citizen, U.S. resident alien, U.S. national, or a resident of Canada or Mexico
The above is simplified; see IRS Publication 501 for full information.
** Worksheet for determining support