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In your case the child lives with the other parent most of the time, the other parent is considered to have full custody for the purposes of the Child Care Benefit. Visit this site for Custody arrangements and your benefits.
You can not put your child as your dependent. If you are “required” to make child support payments for a child and the other parent is not required to, you cannot claim the Amount for an Eligible Dependant for that child. Only the parent who does not pay child support can claim the amount.
Find out more about: Claiming the Amount for an Eligible Dependant Credit.