Usage patterns
〈場所〉の覇を争うto vie for supremacy over (a region, field, etc.)
The two great powers vied for supremacy over Asia.
〈場所〉の覇を求めるto seek dominance over (a region, field, etc.)
The warlord kept fighting in pursuit of dominance over the land.
Expressions
覇を唱えるreign supreme; assume leadership; dominate