My child earned about $10K this year and will file a tax return. Can I also claim him on my return since he still lives with me and I provide support?

How old?

There are a number of factors to consider.

If the child is still in school (high school, college, graduate school), the parent can normally claim the child's exemption if he/she provides more than half of the child's support, until age 25.

If the child is under age 19 at the end of the year, or under age 24 at the end of the year and a full-time student, then the two requirements are:
Child must live with you for more than half of the year
You must provide more than 50% of his support.
It does not matter what he earned.

If child is more than 19/24 yr of age, then NO.

Agreed! MukatA said it better than I did.