If the child is a minor (under the age of 18) or the mother has co-signed for the car then yes, she can take back the car. A child who is irresponsible should not be driving.
most states require car dealers to offer you a limited warranty on the engine and powertrain unless you purchase the vehicle "as is" these warranties are usually at least 30 days or 1,000 miles...
Generally, if the parents are unmarried the mother has sole custody and control in most states until the father can establish his paternity. Remember, a child's mother can always be identified by...
You need to petition the court with jurisdiction and convince the judge that the child would be better off with you. Since you lost custody in the past you will need to provide proof the problems...