"Are you allowed to claim both credits"

Are you confirming they qualify:

"You may be able to claim the child and dependent care credit if you paid expenses for the care of a qualifying individual to enable you (and your spouse, if filing a joint return) to work or actively look for work."

https://www.irs.gov/taxtopics/tc602

You don't get to just Take it. You qualify for it.

Child Tax Credit = the one you get, because you have a kid:

https://www.nerdwallet.com/blog/taxes/qualify-child-child-care-tax-credit/