Continuing the investigation of [1], [2], [3] and [10], we prove here some commutativity theorems for $s$-unital rings $R$ satisfying the polynomial identity $x^t[x^n,y]y^{t'} =\pm x^{s'}[x,y^m]y^s$, resp. $x^t[x^n,y]y^{t'} =\pm y^s[x,y^m]x^{s'}$, where $m,n,s,s',t$ and $t'$ are given non-negative integers such that $m>0$ or $n>0$ and $t+n\ne s'+1$ or $m+s\ne t'+1$ for $m=n$. The additional assumption in these theorems concern some torsion freeness of commutators in$R$.