Despotically tells you how someone rules, decides, or treats other people: in the manner of a despot — cruelly, unfairly, and with all the power in one pair of hands. It often sits next to verbs like rule, govern, control, or run (a country, a company, a classroom). Like despotic, it is formal and a little heavy; in everyday talk people are more likely to say like a dictator or in a really controlling way. When it is used outside politics, it is a sharp complaint, not a mild one.