Summary: Let \(G_H^k(\alpha;r)\) denote the subclasses of normalized harmonic mappings \(f=h+\bar{g}\) in the unit disk \(D\) satisfying the condition \(\text{Re}((1-\alpha)\frac{h(z)}{z}+\alpha h'(z))>|(1-\alpha)\frac{g(z)}{z}+\alpha g'(z)|\) in \(|z|<r,r\in (0,1]\), where \(h'(0)=1,g'(0)=h''(0)=\cdots =h^{(k)}(0)=g^{(k)}(0)=0\) and \(\alpha \geq 0\). In this paper, we first provide the sharp coefficient estimates and the sharp growth theorems for harmonic mappings in the class \(G_H^k(\alpha ;1)\). Next, we derive the geometric properties of harmonic mappings in \(G_H^1(\alpha ; 1)\). Then we study several properties of the sections of \(f\in G_H^k(\alpha ;1)\). Finally, we show that if \(f\in P_H^0(\alpha)\) and \(F\in G_H^1(\beta ;1)\), then the harmonic convolution \(f*F\) is univalent and close-to-convex harmonic function in the unit disk for \(\alpha \in [\frac{1}{2},1),\ \beta \geq 0\).