"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/