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We formulate the multi-band kp theory of hyperfine interactions forsemiconductor nanostructures in the envelope function approximation. We applythis theoretical description to the fluctuations of the longitudinal andtransverse Overhauser field experienced by a hole for a range of InGaAs quantumdots of various compositions and geometries. We find that for a wide range ofvalues of $d$-shell admixture to atomic states forming the top of the valenceband, the transverse Overhauser field caused by this admixture is of the sameorder of magnitude as the longitudinal one, and band mixing adds only a minorcorrection to this result. In consequence, the kp results are well reproducedby a simple box model with the effective number of ions determined by the wavefunction participation number, as long as the hole is confined in thecompositionally uniform volume of the dot, which holds in a wide range ofparameters, excluding very flat dots.