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Yes in theory, the King Charles III and/or his representative the Governor General, could remove the Prime Minister, though the Governor General is chosen following advice (i.e. choice) of the Prime Minister, so I can't imagine that happening in practice.

There has been an instance of a Governor General dismissing a Prime minister.

https://en.wikipedia.org/wiki/1975_Australian_constitutional...

The US president is head of the executive branch, one co-equal third of the entire government, also including the courts (article III of the constitution), and congress (article I).

Speaking for, and possibly misunderstanding, the situation in Ireland:

> But you have a head of state

Yes! Our president.

> who can, in theory, remove the head of government at their own discretion

Ah. No.

A king could… but that’s never going to happen.