In this paper we complete the description of Mergelyan sets for the harmonic Hardy spaceh∞(B) of bounded harmonic functions in the unit ballBof, initiated in [10]. We also characterize those relatively closed subsetsXof a bounded open set ω in the complex plane such that any harmonic functionfon ω can be approximated uniformly on compact subsets of ω by harmonic polynomials and, simultaneously, the same sequence of polynomials converges tofuniformly onxor in Lipschitz norm onXwhenever, respectively, the restriction offtoXis uniformly continuous, or is in lip(αX), 0 > α > 1/2.