custody battle; custody dispute
Refers to a legal or personal conflict over child custody, typically in divorce or separation cases.
The custody battle intensified with the divorce.
After the custody dispute, the mother was awarded custody.
養育権 is 'custody' in the sense of the right to raise a child, often used in legal contexts; 親権争い is the broader battle that may include 養育権.
Compound of 親権 (しんけん, 'parental authority') and 争い (あらそい, 'dispute, fight').