Tax Cuts make people work harder.When you work you are really swapping one valuable resource (time) for another (money). Cutting taxes (which is the equivalent of a pay raise) could mean the person spends more time to get much more money. It could also mean they spend less time to get the same amount of money. It depends on their preferences. As Pettinger implies by his explanation this is an empirical question, not a fallacy.
A Current Account deficit [also known as the trade deficit] doesn’t matter.That happens to be true. Many of his explanations for why the trade deficit is a problem happen to have logical fallacies, though. Concerns of an "unbalanced economy" are inconsistent with a lack of concern about trade flows between cities; concerns of capital flows ignore that the current account is the mirror image of the capital flow, by definition; concerns that a current account deficit means more foreign liabilities ignores that the trade deficit is not debt (when two people trade, no debt is created). That Pettinger acknowledges "some economists" don't think the trade deficit matters demonstrates that this is not a fallacy.
Tax Cuts will boost the Economy.The reasoning behind this supposed fallacy is that of increasing consumption, a straw man since in the strict mathematical sense this won't change anything. (Government spends less and consumption plus investment increases by an equal amount.) The logic behind the argument is in how the funds are spent. Money in the hands of private citizen is more likely to be used more efficiently than money in the hands of political agents (for the normal incentive reasons). Thus tax cuts could easily better efficiency and thus the economy. Granted, Pettinger is correct that borrowing in response to a tax cut does little (it only shifts the spending burden to later generations while crowding out the investments today) but the other way to balance of the budget, cutting spending, won't have such adverse effects.
Popular opinions of economics are filled with blatant nonsense. There are so many, we don't need to the lower the bar and paint differing opinions as something so flawed as a fallacy.