Despite extensive experimental and theoretical efforts, the important issue of the effects of surface magnetic impurities on the topological surface state of a topological insulator (TI) remains unresolved. We elucidate the effects of Cr impurities on epitaxial thin films of (Bi$_{0.5}$Sb$_{0.5}$)$_{2}$Te$_{3}$: Cr adatoms are incrementally deposited onto the TI held in ultrahigh vacuum at low temperatures, and textit{in situ} magnetoconductivity and Hall effect measurements are performed at each increment with electrostatic gating. In the experimentally identified surface transport regime, the measured minimum electron density shows a non-monotonic evolution with the Cr density ($n_{mathrm{Cr}}$): it first increases and then decreases with $n_{mathrm{Cr}}$. This unusual behavior is ascribed to the dual roles of the Cr as ionized impurities and electron donors, having competing effects of enhancing and decreasing the electronic inhomogeneities in the surface state at low and high $n_{mathrm{Cr}}$ respectively. The magnetoconductivity is obtained for different $n_{mathrm{Cr}}$ on one and the same sample, which yields clear evidence that the weak antilocalization effect persists and the surface state remains gapless up to the highest $n_{mathrm{Cr}}$, contrary to the expectation that the deposited Cr should break the time reversal symmetry and induce a gap opening at the Dirac point.
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