Hi seeyadad
Yes, you can apply for Tax Credits. Anyone can apply for tax credits whether they have children or not, you don't have to have children to be entitled to TC. It depends on your income
http://www.taxcredits.inlandrevenue.gov.uk/Hom eNew.aspx
To get Child Element of TC you must of course have children and I believe those children must live with you. The child element of TC is paid, if you are entitled to it, from the birth of the child at a higher rate for the first year and dropping once the child reaches age 1.
For legal purposes and for claiming Child TC's, then main carer is assumed to be the parent or guardian who has the child living with them full-time. Of course it doesn't mean who cares most for the child - in your case or any other case where one parent uses the child against the other, the parent that does this definitely does not care nor have the child's best interests at heart
ojx