Can a non custodial parent claim a child for the child tax credit when they only have them maybe 4 days a month and do not pay for that child

Nope. Only the custodial parent gets that deduction.

I was never married to my sons dad he does not give me any money and he takes his son overnight maybe one night a week if he feels like it

Where can I find a legal answer to that question so that I can give it to him so he will know? Or what happens if he tries to claim him and I already have?

No it is a federal law. You can call the IRS like I did, they did not ask my name. He can not claim them. If he does he will be in violation, which is bad for him.

Mandij,
Use this link I found on the net-- http://www.irs.gov/pub/irs-pdf/p972.pdf It's a pdf file from the IRS. If you look on page 2 it shows the definition of a qualifying child. The child must reside with the one taking advantage of the write off for more than half the year.

You can speak with any public accountant who does taxes.
They will have the legal answer you need, right down to the state laws.

Many of them will give you a quick "yay or nay" for free over the counter.

What country do you live in?

First you need to go to court if you have not, and get a custody order and a order for child support and a order of when he can have visitation.

As for as claiming them as a dependent on your taxes, assuming this is in the US, this is one of the issues normally listed in the custody agreement.
Normally the custodial parent is the one to claim them on their income tax.

Now there are 1000's of parents that do file illegally and then the other parent has to file a complaint with the IRS.
And if it his problem not yours if he files illegally.

The people that do your taxes can explain this to you.

The only way a noncustodial parent can claim a child for the child tax credit, would be if the custodial parent agrees and signs a form 8332 which is a release of exemption and child tax credit. You may not claim Earned Income Credit or anything else, because the child did not live with you more than 6 months.