Not if you meet the HOH rules. Which are that the child lived with you and your wife did not live in your home at any time (not even one night) for the last 6 months of the tax year.
Head of Household is for UNMARRIED taxpayers with a related dependent or married and have not lived with their spouse at anytime during the last 6 months of the tax year AND has a child, stepchild or foster child that can be a dependent.
You may be able to file as head of household if you meet all the following requirements.
1. You are unmarried or “considered unmarried” on the last day of the year.
(You could be considered unmarried if your spouse did not live in your home at any time during the last 6 months of the tax year).
If you were considered married for part of the year and lived in a community property state, special rules may apply in determining your income and expenses. See Publication 555 for more information.
2. You paid more than half the cost of keeping up a home for the year.
3. A “qualifying person” lived with you in the home for more than half the year (except for temporary absences such as school) - a parent does not have to live with you to be a qualifying person.
4. If the qualifying person is your qualifying relative, their gross income must have been less than $4,200 (do not include non taxable Social Security) and you provided more than 1/2 of their support
5. You must be able to claim the dependent for the qualifying person except in the case of divorced or separated parents (that lived apart) and the noncustodial parent is claiming the dependent.
A Qualifying person is either:
A qualifying child or a qualifying closely related relative and meets certain other requirements, however if you are considered unmarried it can only be your child, stepchild, or foster child.
See IRS Publication 501 for more information about who is a qualifying person and a worksheet to determine the cost of keeping up a home.
See IRS Pub 501 for more information
if you qualify to file as HOH you do not qualify to file a joint return or even married filing separate. if you don't your choices are married filing joint or married filing separate
Why do you WANT to file HOH instead of filing a joint return? Did your spouse have income of her own? Is she filing married filing separately? You say she was staying with her mother to take care of her mother. Were you literally apart for at least the last six months of 2019? Who did your child live with?