That country is trying to seize hegemony over Asia.
That country is trying to seize regional hegemony.
覇権争いが激化している。
The struggle for supremacy is intensifying.
The two great powers are competing for supremacy.
The empire established hegemony over the region.
They are maintaining their hegemony through economic power.
支配権 is the concrete 'right of control' over something, whereas 覇権 implies a hegemonic status achieved through overall power and influence.
覇権 is the modern, analytical term for political or international hegemony, while 天下 is the traditional word for holding supreme power over a realm.
覇権 specifically denotes political or international hegemony, while 覇 alone is a more abstract, literary word that can also mean a sports championship.
制覇 means 'conquest' or 'winning a championship' (an action/achievement), while 覇権 is the resulting state of dominance itself.
覇気 means personal 'ambition' or 'spirit to succeed', not the political/military dominance denoted by 覇権.
Sino-Japanese compound of 覇 (supremacy, rule by might) and 権 (power, authority).