  Let $\mathcal{H}$ denote the class of harmonic functions $f$ in $\mathbb{D}:=
\{z\in \mathbb{C}:|z| < 1\}$ normalized by $f(0) = 0 = f_z(0) -1$. For $\alpha
\geq 0$, we consider the following class
$$\mathcal{W}^0_{\mathcal{H}}(\alpha):= \{f = h + \overline{g}\in\mathcal{H}:
{\rm Re\,}(h'(z) + \alpha z h''(z)) >|g'(z) + \alpha z g''(z)|, \quad z\in
\mathbb{D}\}. $$ In this paper, we first prove the coefficient conjecture of
Clunie and Sheil-Small for functions in the class
$\mathcal{W}^0_{\mathcal{H}}(\alpha)$. We also prove growth theorem,
convolution, convex combination properties for functions in the class
$\mathcal{W}^0_{\mathcal{H}}(\alpha)$. Finally, we determine the value of $r$
so that the partial sums of functions in the class
$\mathcal{W}^0_{\mathcal{H}}(\alpha)$ are close-to-convex in $|z|<r$.
