The matrix equation $XA-AX=f(X)$ when $A$ is diagonalizable

$K$ is an algebraically closed field with characteristic $0$ and $f$ is a polynomial or a holomorphic function. We study all solutions of the equation $XA-AX=f(X)$, in the unknown $X\in M_n(K)$, when $A\in M_n(K)$ is diagonalizable.