Head of household requires that you claim your child as a dependent, and that they live in your home more than half the year. If they are 19 or older and not a full time student, you can't claim them as a dependent if they have more than $4700 of taxable income (from any source), and if you can't claim them as a dependent, they don't qualify you to claim head of household.
There is one slight exception: if your child is permanently disabled and unable to perform gainful work, they can be your dependent even if their income is more than $4700. However, being able to work at a minimum wage job is usually considered proof that a person is able to perform gainful work and therefore not considered disabled for tax purposes. (A person can have a medical disability, but if they can work, they aren't disabled for income tax purposes.) The exception is that income earned in a sheltered workshop (like rehab or occupational therapy) doesn't count against them being considered disabled. In the unlikely situation where your child is disabled and unable to perform gainful employment, but has some income from a sheltered workshop, you might still be able to claim them as a dependent even though they are over 19 and have a small job.
