She can relinquish the dependency to him, but he cannot file as Head of Household because the child did not live with him more than half the year. She can file as Head of Household if she otherwise meets the requirements by entering the child on her tax return and claiming him as a qualifying child. Here is how it works with divorced or separated parents:
For divorced or separated parents or parents who live apart, the custodial parent, if eligible, or other eligible person who the child lived with for more than half the year (183 nights in 2019), can claim head of household filing status, the credit for child and dependent care expenses, the exclusion for dependent care benefits, and the earned income credit. The non-custodial parent, if allowed by divorce decree or consent of the custodial parent on form 8332 or similar signed statement, can claim the dependency and child tax credit. For post-2008 divorce decrees or agreements, form 8332 or similar signed statement is required. The child tax benefits cannot be split any other way.
