Mastermind was already mentioned in the question, but the link points only to Knuth's analysis, which assumes that the code is chosen uniformly at random. It is more realistic to model Mastermind as a two-player zero-sum game, and then the natural problem is to compute the equilibrium strategy for both players. This version of the problem has been studied by Bonnet and Viennot, Nash equilibrium in Mastermind, 9th International Conference on Computers and Games, 2016.

There is also a recent paper by Li and Zhu, Playing Mastermind with Wordle-like Feedback (arXiv link), which studies a variant of Mastermind called "Clear Mastermind," which is scored the way the game Wordle is scored. Li and Zhu do not analyze Clear Mastermind game-theoretically, but they do prove some interesting mathematical results.