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I don't see any trope for a state simply having a regent or a character acting as regent. There are these, which are more specific types of regents:
Regent for Life: A nefarious regent who plots to keep their power rather than handing it off to the rightful ruler or heir when the regency is no longer needed.
Evil Chancellor: Redirects from Evil Regent.
The Good Chancellor: Moral opposite to the one above.
Someone who's role it is to be the acting ruler until the real ruler is able to, be it that they're incapacitated, busy overseas or too young to assume direct control