Could you help me to locate a reference in the literature to the following fact: if you act by the Mullineux involution $M$ on a partition $Q$ which is a union of a regular partition $P$ with an added good node of residue $i$, the result will be the Mullineux involution $M(P)$ applied to $P$ to which we add a good node with residue $-i$?



Reference for the action of the Mullineux involution on a partition with an added good node
The equivalence of Kleshchev's algorithm and Mullineux's algorithm was proved by Ford and Kleshchev, but the result they prove is slightly weaker than you want. The result you're asking for is Corollary 4.12 in Bessenrodt & Olsson 'On Residue Symbols and the Mullineux Conjecture', J. Algebraic Combin 7.


