Economic calculation is the only method to determine both the goods higher-valued by people-at-large in society and the lower-cost methods of production for each of those goods. In a market economy, entrepreneurs can determine higher-valued goods by the demands people-at-large have for them and the lower-cost methods of production by the demands competing entrepreneurs-at-large have for the resources.
In central planning, government officials make all production decisions both what goods to produce and what methods of production to use. They cannot determine what goods are higher-valued to people-at-large, but they can (as Mises conceded) supplant their own preferences for those of people-at-large in selecting the goods to be produced. They cannot, however, determine the lower-cost methods of production even for the goods they themselves select to produce. But the calculation problem, per se, does apply to both the selection of higher-valued goods by people-at-large and the selection of lower-cost production methods for whatever goods are chosen.
In a mixed economy, government officials make production decisions for state-run enterprises. If government officials do not sell the goods produced in free competition with private enterprise, then they cannot make accurate decision about what goods are higher-valued to people-at-large. To the extent that they buy inputs in free competition with private enterprise, then the inefficiency of state-run enterprises relative to private enterprises will be manifest. For example, the cost per student in private v. government schools or private v. government healthcare. The coercive power of the state permits state-run enterprises to violate economic calculation and survive even though inefficient. Economic calculation has some impact on decisions of government officials in state-run enterprises, but it does not function as the strict limit imposed on private enterprises.
Take a look at Ludwig von Mises, Human Action on the mixed economy: