Complex sentence: The most important Cabinet post after that of the Premier (who is officially known as President of the Executive Council) is Minister of Finance, followed by the health portfolio, which has a vast budget and is of central political import.
Split into simpler sentences: The most important Cabinet post after that of the Premier (who is officially known as President of the Executive Council) is Minister of Finance. The next most powerful position is the health portfolio which has a vast budget and is of central political import.