We study the dynamics of the spin system that consist of a positively charged II-VI semiconductor quantum dot doped with a single Cr$^+$ ion. The resonant photoluminescence (PL) of the positively charged exciton coupled with the Cr$^+$ spin is used to analyze the main spin relaxation channels. The intensity of the resonant PL is reduced by an optical pumping of the spin of the resident hole-Cr$^+$ complex that can be seen as a nano-magnet. The spin memory can be partially erased by a non-resonant optical excitation. This leads to an increase of the resonant PL signal. The resonant PL is co-circularly polarized and corresponds to relaxation channels that conserve the Cr$^+$ spin $\vert S_z \vert$. The observation in the resonant-PL excitation spectra of optical transitions with a change of the Cr$^+$ spin permits to determine the magnetic anisotropy of the magnetic atom. Optical pumping, auto-correlation measurements and the power dependence of the PL intensity distribution show that the effective temperature of the hole-Cr$^+$ spin system is affected by the optical excitation through the local generation of phonons.
