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Misfires are always worse when the components get hot, why? the resistance goes down as the temperature rises. You need to run a fault code test to see what cylinder is misfiring, it will set a code and you should have a check engine light on.
sounds like the low coolant warning! the light will go out when coolant warms up as it expands and the level will be raised enough to switch off the level sensor may be simple as that check the level when cold.