If you lived with your children for over 6 months during the year and you provided over one-half of the support of the household you would qualify to claim the Head of Household filing status. If you did not live with the children you can not file as Head of Household.
If your children lived with their grandmother for over 6 months during the year and she provided over one-half of the support of the household she would qualify to claim the Head of Household filing status.
She will be affected if you claim the Head of Household filing status when is the one qualified to claim the filing status. Both of you cannot claim the status for the same dependents.
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I think you meant that you and the kids all live at another location which you pay for and you use gma's address for mailing purposes only ... if that is right your using her address on your HOH return will not hurt anyone in any way.