Why does my heat pump blow cold air when it starts/or runs?

This occurs because the heat pump and the indoor blower fan start at the same time and the heat pump requires a few seconds or minutes to build up the required heat with the refrigerant.
Heat Pumps are designed to gradually rise to a desired temperature over a long period of Time with a small heat output and a large amount of air flow. A comfortable temperature could be considered 22°C, yet our body temperature is 37°C; because the Heat Pump is producing air potentially colder than our body temperature the outputted air could seem lukewarm or
even slightly cold to the touch. Don’t worry; this is part of its normal operation of being efficient and producing proper comfort.