Matrices with a diagonalizable power

Since $A$ and $A^{p}$ commute, $A$ stabilizes the eigenspaces of $A^{p}.$ If $0$ is an eigenvalue of $A^{p},$ then the hypothesis on $\mathrm{Ker}(A)$ ensures $A$ maps vectors of $\mathrm{Ker}(A^{p})$ to zero. If $\lambda$ is a nonzero eigenvalue of $A^{p},$ then the restriction $A_{\lambda}$ of $A$ on the corresponding eigenspace of $A^{p}$ has $X^{p}-\lambda$ as an annihilator, a polynomial with complex roots of multiplicity $1,$ hence $A_{\lambda}$ is diagonalizable. We conclude by noting $\C^{n}$ decomposes into eigenspaces of $A^{p}$ by diagonalizability of $A^{p}.$