This paper focuses on nonlinear sub-elliptic systems with drift terms in divergence form, under Dini continuity conditions, where the growth rate satisfies $\frac{2Q}{Q+2}<m<2$, and $Q$ represents the homogeneous dimension in the Heisenberg group. By generalizing the $\mathcal{A}$-harmonic approximation technique to accommodate sub-quadratic growth, we establish the $C^1$ regularity associated with the horizontal gradient of weak solutions away from a negligible set.