We study one-dimensional anisotropic XY-Heisenberg spin-$\frac{1}{2}$ chain with weak random fields $h_i^z S^z_i$ by means of Jordan-Wigner transformation to spinless Luttinger liquid with disorder and bosonization technique. First we investigate phase diagram of the system in terms of dimensionless disorder $\gamma = \left/J^2 \ll 1$ and anisotropy parameter $\Delta = J_z/J_{xy}$ and find the range of these parameters where disorder is irrelevant in the infrared limit and spin-spin correlations are described by power laws. Then we use the diagram technique in terms of plasmon excitations to study low-temperature behavior of heat conductivity $\kappa$ and spin conductivity $\sigma$ in this power-law phase. The obtained Lorentz number $L \equiv \kappa/\sigma T$ differs from the value derived earlier by means of memory function method. We argue also that in the studied region inelastic scattering is strong enough to suppress quantum interference in the low-temperature limit.
