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A qualifying person used for head of household purposes can only be used once for that specific person. If it is a child, the child generally must also be a dependent, or eligible to be a dependent, and must live in the same household for more than 1/2 the year. A child could not meet this requirement for more than one person, and you cannot have two heads of household at the same address, since only one can pay more than 1/2 the cost of support for the household.
Note that there are certain times when head of household and other benefits may be split between two taxpayers for the same child, such as when the noncustodial parent claims the child as a dependent, and the custodial parent claims the child for EIC and head of household purposes.
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The noncustodial parent will absolutely not be allowed to file HOH using this child, since the child did not live with him. If he has another person that qualifies him for HOH status, he may use that person.
The mother is not required to use the HOH status, she may use single, but it is generally not in her advantage to do that if she did pay more than half the cost of the household that the child lives in.