  Based on improving the classical Bohr inequality, we get in this paper some
refined versions for a quasi-subordination family of functions, one of which is
key to build our results. By means of these investigations, for a family of
harmonic mappings defined in the unit disk $\D$, we establish an improved Bohr
inequality with refined Bohr radius under particular conditions. Along the line
of extremal problems concerning the refined Bohr radius, we derive a series of
results. % in a logical way. Here the family of harmonic mappings have the form
$f=h+\overline{g}$, where $g(0)=0$, the analytic part $h$ is bounded by 1 and
that $|g'(z)|\leq k|h'(z)|$ in $\D$ and for some $k\in[0,1]$.
